7.4. The Primary Trigonometric Ratios. LEARN ABOUT the Math. Connecting an angle to the ratios of the sides in a right triangle. Tip.

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1 The Primary Trigonometric Ratios GOL Determine the values of the sine, cosine, and tangent ratios for a specific acute angle in a right triangle. LERN OUT the Math Nadia wants to know the slope of a ski hill. Her trail map shows that the hill makes an angle of 18 with the horizontal. Her friend suggests that she use trigonometry to calculate the slope.? How can the 18 angle be used to determine the slope of the ski hill? trigonometry the branch of mathematics that deals with the properties of triangles and calculations based on these properties EXMPLE 1 onnecting an angle to the ratios of the sides in a right triangle Use the angle of the ski hill to determine the slope of the ski hill. Nadia's Solution ommunication Tip Opposite and sides are named relative to a specific acute angle in a triangle. tangent the ratio of the length of the side to the length of the side for either acute angle in a right triangle; the abbreviation for tangent is tan When using a scientific calculator to calculate the tangent ratio, first make sure that you are in degree mode. 18 run 18 tan = tan 18 = rise run tan rise The slope of the ski hill is approimately I drew a diagram to represent this situation. rise I knew that the ratio is the run same in any right triangle with an 18 angle. So I divided the length of the side to the 18 angle by the length of the side to it. I knew that I could describe the ratio using tangent. I used my calculator to determine tan The Primary Trigonometric Ratios NEL

2 Reflecting. How did Nadia use the properties of similar triangles when she used the tangent ratio to solve the problem?. Why did Nadia use the tangent of the 18 angle instead of the 72 angle to solve the problem? PPLY the Math EXMPLE 2 onnecting each trigonometric ratio to an acute angle Determine the values of the tangent, sine, and cosine ratios for and to four decimal places. Paula s Solution 23 c a 67 b First, I labelled each side using the lower-case letter that matched the angle the side. sine the ratio of the length of the side to the length of the for either acute angle in a right triangle; the abbreviation for sine is sin cosine the ratio of the length of the side to the length of the for either acute angle in a right triangle; the abbreviation for cosine is cos c a 67 sin = = a c sin cos = = b c cos tan = = a b tan b To four decimal places, sin is 0.95, cos is , and tan is I named the sides relative to as,, and. I determined the three primary trigonometric ratios for. primary trigonometric ratios the basic ratios of trigonometry (sine, cosine, and tangent) When using a scientific calculator to calculate the primary trigonometric ratios, press SIN, OS, or TN and then enter the angle, or enter the angle and then press SIN, OS, or TN, depending on your calculator. NEL hapter 7 395

3 b c 23 a I renamed the sides relative to as,, and. ommunication Tip When necessary, trigonometric ratios are usually epressed with four decimal places of accuracy. This is done to calculate the angles with enough precision. arrying as many digits as possible until the final step in a calculation reduces the possibility of variations in answers due to rounding. sin = = b c sin cos = = a c cos tan = = b a tan To four decimal places, sin is , cos is 0.95, and tan is I determined the three primary trigonometric ratios for. EXMPLE 3 onnecting an acute angle to the sides in a right triangle inverse the reverse of an original statement; for eample, if = sin u, the inverse is u = sin -1 When using a scientific calculator to calculate the inverse sine, press 2ND SIN and enter the ratio, or enter the ratio and press 2ND SIN. Determine the measure of u to the nearest degree, using each primary trigonometric ratio cm 2.00 cm Jim s Solution sin u = sin u = u = sin -1 a b u 21.8 u = = = = = I started by determining u using the sine ratio. If I knew the 2.00 angle, its sine would be Since I knew the sine and not the angle, I had to use the inverse of sine, which is sin -1. Since the tangent and cosine ratios both involve the side,, I calculated the length of this side using the Pythagorean theorem. 396 The Primary Trigonometric Ratios NEL

4 cos u = cos u = u = cos -1 a b u 21.6 u 22 To determine the angle using the cosine ratio, I had to use the inverse of cosine, cos -1. My answer is the same as the one I calculated using sine. For help using a TI-83/84 graphing calculator to calculate trigonometric ratios and determine angles, see ppendi -12. If you are using a TI-nspire, see ppendi -48. tan u = tan u = u = tan -1 a b u 21.8 u 22 To determine the angle using the tangent ratio, I had to use the inverse of tangent, tan -1. ll the primary trigonometric ratios gave me the same answer. Net time, I know that I only have to use one of them to determine u. In Summary Key Ideas The primary trigonometric ratios for are sin, cos, and tan. If is one of the acute angles in a right triangle, the primary trigonometric ratios can be determined using the ratios of the sides. Need to Know If is one of the acute angles in a right triangle, the three primary trigonometric ratios for can be written as sin = cos = tan = Using the Pythagorean theorem, = 2 in any right triangle. NEL hapter 7 397

5 HEK Your Understanding 1. a) Which side is to? b) Which side is to? c) Which side is the? 2. Determine the value of each ratio to four decimal places. a) tan 34 b) sin 78 c) cos 49 d) 3. Determine the measure of u to the nearest degree. a) sin u = 0.5 c) cos u = 0.5 b) tan u = 1 d) sin u = sin 12 PRTISING 4. ^XYZ is a right triangle, with X = 90. a) Sketch ^XYZ. Label the sides using lower-case letters. b) Write the ratios for sin Y, cos Y, and tan Y in terms of, y, and z. 5. Determine each ratio, and write it as a decimal to four decimal places. a) sin d) tan b) cos e) cos 15 cm c) tan f) sin 8 cm 6. Decide whether each statement is true or false. Justify your decision. a) sin u = 0.4 b) tan a = 2 c) cos a d) cos u m 17 cm 5.6 m 5.0 m y 7. a) alculate the measure of in the diagram at the left to the nearest degree, using one of the primary trigonometric ratios. b) Do you need to use a primary trigonometric ratio to determine the measure of y? Eplain. 8. Solve for, and epress your answer to one decimal place. a) cos 45 = c) tan 75 = e) cos 60 = 15 6 b) sin 62 = d) tan 80 = 12 f ) sin 45 = Identify the primary trigonometric ratio for u that is equal to each K ratio for the triangle at the left. 50 a) b) c) The Primary Trigonometric Ratios NEL

6 10. Determine the length of. Then state the primary trigonometric ratios for u. a) b) 12 cm 8 cm 13 cm 15 cm 11. Determine the measure of u, to one decimal place, for each triangle in question Determine the measure of u, to one decimal place. a) sin u = 2 5 c) b) cos u = 4 9 d) tan u = 3 sin u = Does cos 60 = 1 mean that the side to the 60 angle 2 measures 1 unit and the measures 2 units? Eplain. 14. Draw two different right triangles for which tan u = 1. Determine T the measurements of all the sides and angles. Then compare the two triangles. 15. a) ould the orange side in ^ at the right be considered an side when determining a trigonometric ratio? Eplain. b) ould the orange side be considered an side when determining a trigonometric ratio? Eplain. c) ould the orange side be considered the when determining a trigonometric ratio? Eplain. Etending 16. For what value of u does sin u = cos u? Include a diagram in your answer. 17. Eplain why the value of tan u increases as the measure of u increases. NEL hapter 7 399