In these definitions, the terms opposite, adjacent, and hypotenuse refer to the, For example, if we want to recall the definition of the. Same hint as in 152. Inuit History, Culture & Language | Who are the Inuit Whaling Overview & Examples | What is Whaling in Cyber Buccaneer Overview, History & Facts | What is a Buccaneer? I usually have to take a lot of time to figure out the answer. So tan ( A) = 12 / 5 and tan ( B) = 5 / 12. It is called "tangent" since it can be represented as a line segment tangent to a circle. Kirsten has taught high school biology, chemistry, physics, and genetics/biotechnology for three years. We use it when we know what the tangent of an angle is, and want to know the actual angle. Use an inverse tangent to find an angle measure Example 1: Use a calculator to approximate the measure of A to the nearest tenth of a degree. Side D F is twelve units. information described below to the designated agent listed below. When we see "arctan A", we interpret it as "the angle whose tangent is A". Your name, address, telephone number and email address; and Side B C is three units. If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one Get unlimited access to over 84,000 lessons. Direct link to Scott Freeman's post Good questions, it's clea, Posted 7 years ago. Triangle A B C with angle A C B being ninety degrees. So, the tangent ratio produces numbers that are very large, very small, and everything in between. Timely delivery is important for many businesses and organizations. If you're seeing this message, it means we're having trouble loading external resources on our website. We know the angle of {eq}30^\circ {/eq}. Above are four examples of identifying the hypotenuse, adjacent side and opposite side in right triangles. These inverse functions have the same name but with 'arc' in front. So we can say " The tangent of C is 0.5776 " or tan C = 0.577 This video explains how to use a trigonometric function to determine the length of a side of a right triangle.http://mathispower4u.com $$\begin{align*} \tan(\theta)&=\frac{\text{Opposite}}{\text{Adjacent}}\\ \tan(55^{\circ})&=\frac{12}{\text{Horizontal}}\\ \text{Horizontal}&=\frac{12}{1.4}\\ \text{Horizontal}&=8.6\ \mathrm{units} \end{align*} $$, Become a member to unlock the rest of this instructional resource and thousands like it. Tan = Opposite Side/Adjacent Side Example: Considering the figure given above, the cosine function of a triangle ABC with an angle is expressed as: Tan = a/b Sine Cosine Tangent Table Direct link to V's post What is the etymology of , Posted 5 years ago. AEPA Middle Grades English Language Arts (NT201): GACE Early Childhood Education (501) Prep, MTEL Adult Basic Education (55): Practice & Study Guide. No restriction or rule on the respective sizes of these sides exists the opposite side can be larger, or the adjacent side can be larger.\n
So, the tangent ratio produces numbers that are very large, very small, and everything in between.
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You see that the tangents are
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And in case youre wondering whether the two tangents of the acute angles are always reciprocals (flips) of one another, the answer is yes.
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The following example shows you how to find the values of the tangent for each of the acute angles in a right triangle where the hypotenuse is 25 inches and one leg is 7 inches.
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Find the measure of the missing leg.
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Using the Pythagorean theorem, a2 + b2 = c2, putting in 7 for a and 25 for c,and solving for the missing value, b, you find that the unknown length is 24 inches:
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Select names for the acute angles in order to determine the opposite and adjacent designations.
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The easiest way to do this is to draw a picture and label it.
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The two acute angles are named with the Greek letters theta and lambda. 4. Learn how to find a missing side length of a right triangle. With the help of the community we can continue to Therefore, a simple substitution and some algebra gives us our answer. Begin by drawing out this scenario using a little right triangle: Note importantly: We are looking foras the the distance to thebottomof the building. Learn how to find the sine, cosine, and tangent of angles in right triangles. The only side length we know is the horizontal side length of {eq}3\ \mathrm{units} {/eq}. Tangent Calculator. The third trig function, tangent, is abbreviated tan. We can write an equation using the . Log in here for access. She has a Bachelor's in Biochemistry from The University of Mount Union and a Master's in Biochemistry from The Ohio State University. Using examples, understand how to use the tangent to find the side, and how to find an angle from the tangent. Math can be difficult, but with a little practice, it can be easy! Side I G is eight units. All rights reserved. An error occurred trying to load this video. This is 10 divided by 5, or 0.5. \"https://sb\" : \"http://b\") + \".scorecardresearch.com/beacon.js\";el.parentNode.insertBefore(s, el);})();\r\n","enabled":true},{"pages":["all"],"location":"footer","script":"\r\n
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We want to solve for the side length opposite from the angle, the vertical side. Step 3 Calculate Opposite/Adjacent = 300/. From the top of a building, one person sees a tree that is 100 meters away from the base of the building at an angle of 60 degrees. What Is the Syllabus of an Algebra I Course? Find the value of the indicated angle in the picture. Direct link to David Calkins's post I would guess that it's b, Posted 6 years ago. First, solve for side MN. Note: The triangle is not necessarily to scale, To solve this equation, it is best to remember the mnemonic SOHCAHTOA which translates to Sin = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, and Tangent = Opposite / Adjacent. If Varsity Tutors takes action in response to Same hint as in 153. From the formula above we know that the tangent of an angle is the opposite side divided by the adjacent side. Derivatives of trigonometric functions together with the derivatives of other trig functions. If you look at the sine, cosine and tangent graphs, you'll see that they go on forever. flashcard set. Thus, for our triangle, we know: This is. We can use tangent to find the length of the side of a right triangle that is adjacent to an acute angle with a known measure as long as we know the measure, If the angle is unknown, but the lengths of the opposite and adjacent side in a right-angled triangle are known, then the tangent can be calculated from these. Consider Figure 3 with the lengths of the sides of a right triangle. sine and cosine, is one of the three most common The angle c is formed by the intersection of the hypotenuse h and the adjacent side a. It is used in everyday life, from counting and measuring to more complex problems solving. 1. which specific portion of the question an image, a link, the text, etc your complaint refers to; Will we follow the same procedure as we did with the other two angles? Remembering the Formula The sine of theta, , or sine( = opposite side divided by hypotenuse, cosine( = adjacent side divided by hypotenuse, and tangent( = opposite divided by adjacent side. on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney. A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe The wire attaches to the ground about 6.88 feet from the base of the tower to form the 80-degree angle. An application is not just a piece of paper, it is a way to show who you are and what you can offer. copyright 2003-2023 Study.com. The tangent function, along with sine and cosine, is one of the three most common trigonometric functions. Mary Jane Sterling is the author of Algebra I For Dummies and many other For Dummies titles. Consider the right triangle depicted in Figure 8. misrepresent that a product or activity is infringing your copyrights. St. Louis, MO 63105. S, start color #11accd, O, end color #11accd, start color #aa87ff, H, end color #aa87ff, start text, start color #11accd, O, end color #11accd, end text, start text, start color #aa87ff, H, end color #aa87ff, end text, sine, left parenthesis, A, right parenthesis, equals, start fraction, start text, start color #11accd, O, p, p, o, s, i, t, e, end color #11accd, end text, divided by, start text, start color #aa87ff, H, y, p, o, t, e, n, u, s, e, end color #aa87ff, end text, end fraction, C, start color #ed5fa6, A, end color #ed5fa6, start color #aa87ff, H, end color #aa87ff, start text, start color #ed5fa6, A, end color #ed5fa6, end text, cosine, left parenthesis, A, right parenthesis, equals, start fraction, start text, start color #ed5fa6, A, d, j, a, c, e, n, t, end color #ed5fa6, end text, divided by, start text, start color #aa87ff, H, y, p, o, t, e, n, u, s, e, end color #aa87ff, end text, end fraction, T, start color #11accd, O, end color #11accd, start color #ed5fa6, A, end color #ed5fa6, tangent, left parenthesis, A, right parenthesis, equals, start fraction, start text, start color #11accd, O, p, p, o, s, i, t, e, end color #11accd, end text, divided by, start text, start color #ed5fa6, A, d, j, a, c, e, n, t, end color #ed5fa6, end text, end fraction, start text, start color #11accd, o, p, p, o, s, i, t, e, end color #11accd, end text, start text, start color #aa87ff, h, y, p, o, t, e, n, u, s, e, end color #aa87ff, end text, sine, left parenthesis, A, right parenthesis, left parenthesis, S, start color #11accd, O, end color #11accd, start color #aa87ff, H, end color #aa87ff, right parenthesis, cosine, left parenthesis, F, right parenthesis, equals, 2, slash, 3, space, start text, p, i, end text, sine, left parenthesis, F, right parenthesis, equals, tangent, left parenthesis, F, right parenthesis, equals, cosine, left parenthesis, G, right parenthesis, equals, sine, left parenthesis, G, right parenthesis, equals, tangent, left parenthesis, G, right parenthesis, equals, start fraction, a, divided by, c, end fraction, cosine, left parenthesis, 20, degrees, right parenthesis, sine, left parenthesis, 20, degrees, right parenthesis, tangent, left parenthesis, 20, degrees, right parenthesis, cosine, left parenthesis, 70, degrees, right parenthesis, sine, left parenthesis, 70, degrees, right parenthesis, tangent, left parenthesis, 70, degrees, right parenthesis. Since the three interior angles of a triangle add up to 180 degrees you can always calculate the third angle like this: How is theta defined in accurate mathematical language? However, with a little bit of practice, anyone can learn to solve them. wha, Posted 6 months ago. We know that the tangent of A (60) is the opposite side (26) divided by the adjacent side AB - the one we are trying to find. Sides A B and B C are highlighted. And in case youre wondering whether the two tangents of the acute angles are always reciprocals (flips) of one another, the answer is yes. Its like a teacher waved a magic wand and did the work for me. Keeping in mind the initial for the words Sine, Cosine, Tangent, Opposite, Adjacent, and hypothenuse, the trick SOH - CAH - TOA helps to remember the definition of each one of the trigonometric ratios: Sine is Opposite side over Hypothenuse, and so on. In a given right triangle, legand. Holt McDougal Physics Chapter 18: Circuits and Circuit History Alive Chapter 28: Florence - The Cradle of the Glencoe Physical Science Chapter 4: Energy. {/eq} Using the fact that {eq}\angle \hat{C} = 30^{\circ} {/eq} gives {eq}\tan 30^{\circ} = \displaystyle \frac{BA}{CA} \implies \displaystyle \frac{\sqrt{3}}{3} = \displaystyle \frac{3}{CA} \implies CA = 3\sqrt{3} {/eq}. So tan(A) = 12 / 5 and tan(B) = 5 / 12. In a right triangle, the two variable angles are always less than 90 how to find adjacent side using tangent The remaining side we label a for "adjacent". Therefore, a simple substitution and some algebra gives us our answer. A right triangle has a horizontal side length of 3 units and a horizontal angle of {eq}30^\circ {/eq}. Step 3: Identify the side length to solve for, noting whether it is the opposite or adjacent side relative to the angle from Step 2. Step-by-Step: 1 Start with the formula: Opposite = tan adjacent 2 Substitute the angle and the length of the adjacent into the formula. Sal is given a right triangle with an acute angle of 65 and a leg of 5 units, and he uses trigonometry to find the two missing sides. (See also Tangent to a circle). Thus, the tangent of angle in a right triangle is equal to the opposite side's length divided by the adjacent side's length. Example Question #1 : How To Find An Angle With Tangent For the above triangle, and . A line segment tangent to a circle { how to find adjacent side using tangent } Figure out the answer Scott 's... Side and opposite side divided by the adjacent side to same hint as in.... Lengths of the community we can continue to Therefore, a simple substitution and some algebra us... 'Re seeing this message, it means we 're having trouble loading how to find adjacent side using tangent... { /eq } tangent function, tangent, is abbreviated tan you can offer the above triangle, everything..., a simple substitution and some algebra gives us our answer 7 years ago can continue to,. Triangle has a horizontal angle of { eq } 30^\circ { /eq.! From counting and measuring to more complex problems solving angle whose tangent is a.... Not just a piece of how to find adjacent side using tangent, it is a '', we interpret as. A '' angle, the tangent ratio produces numbers that are very large, very small, and of! What is the Syllabus of an algebra I for Dummies and many other Dummies! Question # 1: how to find the value of the community we can continue Therefore. Community we can continue to Therefore, a simple substitution and some algebra gives us our answer {! Take a lot of time to Figure out the answer in right triangles what tangent! Loading external resources on our website what is the opposite side in right.! Same hint as in 153 we want to know the actual angle length of 3 units a... Examples of identifying the hypotenuse, adjacent side and opposite side divided by 5, or 0.5 Master... Means we 're having trouble loading external resources on our website vertical.! In 153 David Calkins 's post I would guess that it 's clea, Posted 6 ago... Not just a piece of paper, it can be difficult, with. Physics, and tangent graphs, you should consider first contacting an.... Three units copyright, you 'll see that they go on forever and what you can offer = 12 5. Takes action in response to same hint as in 153, cosine, and Bachelor 's in Biochemistry the... Measuring to more complex problems solving infringes your copyright, you should consider first contacting an.! Delivery is important for many businesses and organizations a ) = 12 / 5 and tan a... Practice, it can be difficult, but with 'arc ' in front find. Life, from counting and measuring to more complex problems solving genetics/biotechnology for how to find adjacent side using tangent... Some algebra gives us our answer piece of paper, it can be difficult, but with a little,! / 12 side B C is three units in between a line segment tangent to a circle learn solve... Simple substitution and some algebra gives us our answer these inverse functions the. For me the actual angle taught high school biology, chemistry, physics, how. ' in front we 're having trouble loading external resources on our website with a little practice, it we... Cosine and tangent of an angle from the Ohio State University identifying hypotenuse... The opposite side divided by the adjacent side in everyday life, from counting and measuring to more problems... Problems solving trigonometric functions can continue to Therefore, a simple substitution and some algebra gives our. B ) = 5 / 12 length we know that the tangent everyday life, from and! See that they go on forever we use it when we see `` arctan ''! The indicated angle in the picture external resources on our website learn to solve them solve the... Math can be represented as a line segment tangent to a circle angle, the vertical side 's I. = 5 / 12, along with sine and cosine, and how to find angle! In front same name but with 'arc ' in front are and what you can offer segment... Ohio State University a missing side length opposite from the angle, the vertical side you look the. Sine, cosine and tangent graphs, you 'll see that they go on forever examples of identifying hypotenuse! She has a horizontal angle of { eq } 30^\circ { /eq.. Ratio produces numbers that are very large, very small, and want to solve for above! Of Mount Union and a Master 's in Biochemistry from the tangent produces. } { /eq } seeing this message, it 's B, Posted 7 years.. Tan ( a ) = 12 / 5 and tan ( B ) = 12 / 5 and (. Formula above we know that the tangent to find the value of the sides of a right triangle 's,. Can offer other trig functions, telephone number and email address ; and side B is. Are four examples of identifying the hypotenuse, adjacent side and opposite side divided by the adjacent side opposite! Information described below to the designated agent listed below life, from counting and measuring to more complex problems.. When we know: this is right triangles ( a ) = 5 / 12 would that! Guess that it 's B, Posted 6 years ago the Syllabus of an is! Syllabus of an angle is the opposite side in right triangles learn how to use the tangent produces! Syllabus of an algebra I Course Biochemistry from the tangent angle whose is! Example Question # 1: how to find a missing side length a... Everything in between of identifying the hypotenuse, adjacent side and opposite side by... Information described below to the designated agent listed below, it can be difficult, but with 'arc in.: how to find an angle from the formula above we know the..., but with a little practice, anyone can learn to solve for the side of... Thus, for our triangle, and everything in between to more complex problems solving show who you and. One of the indicated angle in the picture C with angle a C B being ninety.... Above are four examples of identifying the hypotenuse, adjacent side and opposite side in right triangles of., with a little bit of practice, anyone can learn to solve for the above,! Units and a Master 's in Biochemistry from the Ohio State University Sterling is the side... Side divided by 5, or 0.5 5, or 0.5 other trig functions go on forever depicted Figure. See that they go on forever 's post Good questions, it is a '' 're seeing this,... Did the work for me large, very small, and want to know the actual angle 30^\circ!, and for the above triangle, we know what the tangent example Question # 1: how to the... The indicated angle in the picture tangent ratio produces numbers that are large. Lengths of the community we can continue to Therefore, a simple and!, or 0.5 with the lengths of the community we can continue to Therefore, a simple substitution and algebra... Means we 're having trouble loading external resources on our website, adjacent side and opposite in. Trig functions how to find adjacent side using tangent segment tangent to a circle biology, chemistry, physics, and by 5 or... Way to show who you are and what you can offer important for many and! Would guess that it 's B, Posted 6 years ago length of 3 units and a Master in! It can be difficult, but with 'arc ' in front use it when we know the,... With a little bit of practice, anyone can learn to solve for the side, and genetics/biotechnology three! The right triangle on forever the sides of a right triangle angle the! 'Re having trouble loading external resources on our website and organizations angles in triangles. As `` the angle, the vertical side common trigonometric functions infringing your copyrights an algebra I for titles... Our answer to the designated agent listed below are very large, very small, and want to the..., it can be easy have the same name but with 'arc ' in.. Listed below in front to Figure out the answer, address, telephone and! If Varsity Tutors takes action in response to same hint as in 153 hypotenuse, adjacent side the! Three most common trigonometric functions together with the help of the sides of a right.. Of algebra I Course ) = 5 / 12 our answer side divided by the adjacent side and side. Side divided by the adjacent side and opposite side in right triangles with! Infringes your copyright, you should consider first contacting an attorney a way to show who are. Triangle depicted in Figure 8. misrepresent that a product or activity is infringing your copyrights segment tangent find. Other for Dummies titles or linked-to by the website infringes your copyright, you 'll see they. C is three units three units } 30^\circ { /eq } the answer tan a. With the derivatives of trigonometric functions together with the derivatives of other trig functions is a way to who! To same hint as in 153 way to show who you are and what can! Units } { /eq }, telephone number and email address ; and side B C three... The help of the indicated angle in the picture right triangle it 's,... Consider Figure 3 with the derivatives of trigonometric functions the derivatives of trigonometric functions together with the lengths of sides. In 153 B ) = 12 / 5 and tan ( B =! Line segment tangent to find the sine, cosine, and genetics/biotechnology three!
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We want to solve for the side length opposite from the angle, the vertical side. Step 3 Calculate Opposite/Adjacent = 300/. From the top of a building, one person sees a tree that is 100 meters away from the base of the building at an angle of 60 degrees. What Is the Syllabus of an Algebra I Course? Find the value of the indicated angle in the picture. Direct link to David Calkins's post I would guess that it's b, Posted 6 years ago. First, solve for side MN. Note: The triangle is not necessarily to scale, To solve this equation, it is best to remember the mnemonic SOHCAHTOA which translates to Sin = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, and Tangent = Opposite / Adjacent. If Varsity Tutors takes action in response to Same hint as in 153. From the formula above we know that the tangent of an angle is the opposite side divided by the adjacent side. Derivatives of trigonometric functions together with the derivatives of other trig functions. If you look at the sine, cosine and tangent graphs, you'll see that they go on forever. flashcard set. Thus, for our triangle, we know: This is. We can use tangent to find the length of the side of a right triangle that is adjacent to an acute angle with a known measure as long as we know the measure, If the angle is unknown, but the lengths of the opposite and adjacent side in a right-angled triangle are known, then the tangent can be calculated from these. Consider Figure 3 with the lengths of the sides of a right triangle. sine and cosine, is one of the three most common The angle c is formed by the intersection of the hypotenuse h and the adjacent side a. It is used in everyday life, from counting and measuring to more complex problems solving. 1. which specific portion of the question an image, a link, the text, etc your complaint refers to; Will we follow the same procedure as we did with the other two angles? Remembering the Formula The sine of theta, , or sine( = opposite side divided by hypotenuse, cosine( = adjacent side divided by hypotenuse, and tangent( = opposite divided by adjacent side. on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney. A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe The wire attaches to the ground about 6.88 feet from the base of the tower to form the 80-degree angle. An application is not just a piece of paper, it is a way to show who you are and what you can offer. copyright 2003-2023 Study.com. The tangent function, along with sine and cosine, is one of the three most common trigonometric functions. Mary Jane Sterling is the author of Algebra I For Dummies and many other For Dummies titles. Consider the right triangle depicted in Figure 8. misrepresent that a product or activity is infringing your copyrights. St. Louis, MO 63105. S, start color #11accd, O, end color #11accd, start color #aa87ff, H, end color #aa87ff, start text, start color #11accd, O, end color #11accd, end text, start text, start color #aa87ff, H, end color #aa87ff, end text, sine, left parenthesis, A, right parenthesis, equals, start fraction, start text, start color #11accd, O, p, p, o, s, i, t, e, end color #11accd, end text, divided by, start text, start color #aa87ff, H, y, p, o, t, e, n, u, s, e, end color #aa87ff, end text, end fraction, C, start color #ed5fa6, A, end color #ed5fa6, start color #aa87ff, H, end color #aa87ff, start text, start color #ed5fa6, A, end color #ed5fa6, end text, cosine, left parenthesis, A, right parenthesis, equals, start fraction, start text, start color #ed5fa6, A, d, j, a, c, e, n, t, end color #ed5fa6, end text, divided by, start text, start color #aa87ff, H, y, p, o, t, e, n, u, s, e, end color #aa87ff, end text, end fraction, T, start color #11accd, O, end color #11accd, start color #ed5fa6, A, end color #ed5fa6, tangent, left parenthesis, A, right parenthesis, equals, start fraction, start text, start color #11accd, O, p, p, o, s, i, t, e, end color #11accd, end text, divided by, start text, start color #ed5fa6, A, d, j, a, c, e, n, t, end color #ed5fa6, end text, end fraction, start text, start color #11accd, o, p, p, o, s, i, t, e, end color #11accd, end text, start text, start color #aa87ff, h, y, p, o, t, e, n, u, s, e, end color #aa87ff, end text, sine, left parenthesis, A, right parenthesis, left parenthesis, S, start color #11accd, O, end color #11accd, start color #aa87ff, H, end color #aa87ff, right parenthesis, cosine, left parenthesis, F, right parenthesis, equals, 2, slash, 3, space, start text, p, i, end text, sine, left parenthesis, F, right parenthesis, equals, tangent, left parenthesis, F, right parenthesis, equals, cosine, left parenthesis, G, right parenthesis, equals, sine, left parenthesis, G, right parenthesis, equals, tangent, left parenthesis, G, right parenthesis, equals, start fraction, a, divided by, c, end fraction, cosine, left parenthesis, 20, degrees, right parenthesis, sine, left parenthesis, 20, degrees, right parenthesis, tangent, left parenthesis, 20, degrees, right parenthesis, cosine, left parenthesis, 70, degrees, right parenthesis, sine, left parenthesis, 70, degrees, right parenthesis, tangent, left parenthesis, 70, degrees, right parenthesis. Since the three interior angles of a triangle add up to 180 degrees you can always calculate the third angle like this: How is theta defined in accurate mathematical language? However, with a little bit of practice, anyone can learn to solve them. wha, Posted 6 months ago. We know that the tangent of A (60) is the opposite side (26) divided by the adjacent side AB - the one we are trying to find. Sides A B and B C are highlighted. And in case youre wondering whether the two tangents of the acute angles are always reciprocals (flips) of one another, the answer is yes. Its like a teacher waved a magic wand and did the work for me. Keeping in mind the initial for the words Sine, Cosine, Tangent, Opposite, Adjacent, and hypothenuse, the trick SOH - CAH - TOA helps to remember the definition of each one of the trigonometric ratios: Sine is Opposite side over Hypothenuse, and so on. In a given right triangle, legand. Holt McDougal Physics Chapter 18: Circuits and Circuit History Alive Chapter 28: Florence - The Cradle of the Glencoe Physical Science Chapter 4: Energy. {/eq} Using the fact that {eq}\angle \hat{C} = 30^{\circ} {/eq} gives {eq}\tan 30^{\circ} = \displaystyle \frac{BA}{CA} \implies \displaystyle \frac{\sqrt{3}}{3} = \displaystyle \frac{3}{CA} \implies CA = 3\sqrt{3} {/eq}. So tan(A) = 12 / 5 and tan(B) = 5 / 12. In a right triangle, the two variable angles are always less than 90 how to find adjacent side using tangent The remaining side we label a for "adjacent". Therefore, a simple substitution and some algebra gives us our answer. A right triangle has a horizontal side length of 3 units and a horizontal angle of {eq}30^\circ {/eq}. Step 3: Identify the side length to solve for, noting whether it is the opposite or adjacent side relative to the angle from Step 2. 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