Trigonometric Ratios. Lori Jordan Kate Dirga. Say Thanks to the Authors Click (No sign in required)

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1 Trigonometric Ratios Lori Jordan Kate Dirga Say Thanks to the Authors Click (No sign in required)

2 To access a customizable version of this book, as well as other interactive content, visit AUTHORS Lori Jordan Kate Dirga CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-source, collaborative, and web-based compilation model, CK-12 pioneers and promotes the creation and distribution of high-quality, adaptive online textbooks that can be mixed, modified and printed (i.e., the FlexBook textbooks). Copyright 2017 CK-12 Foundation, The names CK-12 and CK12 and associated logos and the terms FlexBook and FlexBook Platform (collectively CK-12 Marks ) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution-Non-Commercial 3.0 Unported (CC BY-NC 3.0) License ( licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the CC License ), which is incorporated herein by this reference. Complete terms can be found at terms-of-use. Printed: October 15, 2017

3 Chapter 1. Trigonometric Ratios CHAPTER 1 Trigonometric Ratios Learning Objectives Here you ll define and apply the trigonometric ratios sine, cosine and tangent to solve for the lengths of unknown sides in right triangles. TEKS M.6D (see actual TEKS in resources) Learning Objectives To learn the trigonometric ratios sine, cosine, and tangent. To apply the ratios to find lengths and angles of right triangles. Vocabulary Sine - The sine is equal to the length of the side opposite the angle θ, divided by the length of the triangle s hypotenuse. Cosine - The cosine is equal to the length of the side adjacent to the angle θ, divided by the length of the triangle s hypotenuse. Tangent - The tangent is equal to the length of the side opposite the angle θ, divided by the length of the side adjacent to the angle θ. Introduction Trigonometry is simply geometrical constructions where the ratio of a triangle s side lengths is used to find the measurement of the angle. Mathematicians have used trigonometry for centuries to accurately determine distances without having to physically measure them. It can also be used to calculate angles that would be very difficult to measure. Trigonometry has uses in such areas as surveying, navigation, drawing and architecture. Trigonometry (from Greek trigōnon, "triangle" and metron, "measure" is a branch of mathematics that observes relationships which involve the angles and lengths of triangles. Trigonometry became known in the Hellenistic world during the third century BC from studies and applications of geometry. The 3rd-century astronomers recognized the patterns of the lengths of the sides of right triangles and the relationship of the angles in the right triangles. Trigonometric functions were defined as these calculations took form. Trigonometry is most simply associated with planar right-angle triangles (each of which is a two-dimensional triangle with one angle equal to a right angle). There is substantial number of uses of trigonometry and trigonometric functions. For instance, the technique of triangulation is used in astronomy to measure the distance to nearby stars, in geography to measure distances between landmarks, and in satellite navigation systems. The sine and cosine functions are fundamental to the theory of periodic functions such as those that describe sound and light waves. Fields that use trigonometry or trigonometric functions include astronomy (especially for locating apparent positions of celestial objects, in which spherical trigonometry is essential) and hence navigation (on the oceans, in aircraft, and in space), music theory, audio synthesis, acoustics, optics, electronics, probability theory, statistics, biology, 1

4 medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, many physical sciences, land surveying and geodesy, architecture, image compression, phonetics, economics, electrical engineering. 1. a right triangle Choose an x-axis and a y-axis (orthonormal) and let O be the origin. A circle of radius one centered at O is called the trigonometric circle or the unit circle. Turning counterclockwise is the positive orientation in trigonometry. Angles are measured starting from the x-axis. The units used to measure an angle are degree and radian. A right angle is an angle whose measure is exactly 90 degrees or pi/2 radians. In this theory, we use mainly radians. Each real number t corresponds to exactly one angle, and to exactly one point P on the unit circle. We call that point the image point of t. 1. a unit circle 2

5 Chapter 1. Trigonometric Ratios The trigonometric ratios sine, cosine and tangent refer to the known ratios between particular sides in a right triangle based on an acute angle measure. In this right triangle, side c is the hypotenuse. If we consider the angle B, then we can describe each of the legs by its position relative to angle B: side a is adjacent to B; side b is opposite B If we consider the angle A, then we can describe each of the legs by its position relative to angle A: side b is adjacent to A; side a is opposite A Now we can define the trigonometry ratios as follows: Sine is opposite Cosine is hypotenuse ad jacent hypotenuse T angent is opposite ad jacent A shorthand way to remember these ratios is to take the letters in red above and write the phrase: SOH CAH TOA Now we can find the trigonometric ratios for each of the acute angles in the triangle above. sina = a c sinb = b c cosa = b c cosb = a c tana = a b tanb = b a It is important to understand that given a particular (acute) angle measure in a right triangle, these ratios are constant no matter how big or small the triangle. For example; if the measure of the angle is 25, then sin and ratio of the opposite side to the hypotenuse is always no matter how big or small the triangle. 3

6 Example A Find the trig ratios for the acute angles R and P in PQR. Solution: From angle R, O = 8; A = 15; and H = 17. Now the trig ratios are: sinr = 8 15 ; cosr = ; tanr = 8 15 From angle P, O = 15; A = 8; and H = 17. Now the trig ratios are: sinp = ; cosp = 8 17 ; tanp = 15 8 Do you notice any patterns or similarities between the trigonometric ratios? The opposite and adjacent sides are switched and the hypotenuse is the same. Notice how this switch affects the ratios: sinr = cosp cosr = sinp tanr = 1 tanp Example B Use trigonometric ratios to find the x and y. Solution: First identify or label the sides with respect to the given acute angle. So, x is opposite, y is hypotenuse (note that it is the hypotenuse because it is the side opposite the right angle, it may be adjacent to the given angle but the hypotenuse cannot be the adjacent side) and 6 is the adjacent side. 4

7 Chapter 1. Trigonometric Ratios To find x, we must use the given length of 6 in our ratio too. So we are using opposite and adjacent. Since tangent is the ratio of opposite over adjacent we get: tan35 = x 6 x = 6tan35 multiply both sides by 6 x 4.20 Use the calculator to evaluate-type in 6TAN(35) ENTER NOTE: make sure that your calculator is in DEGREE mode. To check, press the MODE button and verify that DEGREE is highlighted (as opposed to RADIAN). If it is not, use the arrow buttons to go to DEGREE and press ENTER. The default mode is radian, so if your calculator is reset or the memory is cleared it will go back to radian mode until you change it. To find y using trig ratios and the given length of 6, we have adjacent and hypotenuse so we ll use cosine: cos35 = 6 y cos35 = 6 1 y set up a proportion to solve for y 6 = ycos35 cross multiply y = 6 cos35 divide bycos35 y = 7.32 Use the calculator to evaluate-type in 6/TAN(35) ENTER Alternatively, we could find y using the value we found for x and the Pythagorean theorem: = y = y 2 y 7.32 The downside of this method is that if we miscalculated our x value, we will double down on our mistake and guarantee an incorrect y value. In general you will help avoid this kind of mistake if you use the given information whenever possible. Example C Given ABC, with m A = 90,m C = 20 and c = 9, find a and b. Solution: Visual learners may find it particularly useful to make a sketch of this triangle and label it with the given information: 5

8 To find a (the hypotenuse) we can use the opposite side and the sine ratio: sin20 = 9 a, solving as we did in Example B we get a = 9 sin To find b (the adjacent side) we can use the opposite side and the tangent ratio: tan20 = 9 b, solving for b we get b = 9 tan Concept Problem Revisit If you draw the triangle described in this problem, you will see that the sine opposite 4 each of the acute angles in the same. It is hypotenuse. So we need to find the hypotenuse. Let s use the Pythagorean Theorem. hypotenuse of = c = c 2 32 = c 2 c = Therefore, the sine of both of the acute angles is or Guided Practice 1. Use trig ratios to find x and y: 2. Given ABC with m B = 90,m A = 43 and a = 7, find b and c. 3. The base of a playground slide is 6 ft from the base of the platform and the slide makes a 60 angle with the ground. To the nearest tenth of a foot, how high is the platform at the top of the slide? Answers 1. For x: 6 cos62 = 5 x x = 5 cos

9 Chapter 1. Trigonometric Ratios For y: tan62 = y 5 y = 5tan For b: For c: sin43 = 7 b b = 7 sin tan43 = 7 c c = 7 tan , so the height of the platform is 10.4 ft tan60 = h 6 h = 6tan Explore More Use you calculator to find the following trigonometric ratios. Give answers to four decimal places. 1. sin35 2. tan72 3. cos48 4. tan45 5. sin30 6. cos88 7. Write the three trigonometric ratios of each of the acute angles in the triangle below. Use trigonometric ratios to find the unknown side lengths in the triangles below. Round your answers to the nearest hundredth. 7

10 For problems use the given information about ABC with right angle B to find the unknown side lengths. Round your answer to the nearest hundredth. 11. a = 12 and m A = m C = 75 and b = c = 7 and m A = A ramp needs to have an angle of elevation no greater than 10 degrees. If the door is 3 ft above the sidewalk level, what is the minimum possible ramp length to the nearest tenth of a foot? A ship, Sea Dancer, is 10 km due East of a lighthouse. A second ship, Nelly, is due north of the lighthouse. A spotter on the Sea Dancer measures the angle between the Nelly and the lighthouse to be 38. How far apart are the two ships to the nearest tenth of a kilometer?

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