9.4 Trigonometry: Cosine Ratio Geometry 9.4 Trigonometry: Cosine Ratio

9.4 Cosine Ratio Cosine Ratio, Secant Ratio, and Inverse Cosine Objectives Use the cosine ratio in a right triangle to solve for unknown side lengths. Use the secant ratio in a right triangle to solve for unknown side lengths. Relate the cosine ratio to the secant ratio. Use the inverse cosine in a right triangle to solve for unknown angle measures. SOH – CAH – TOA

9.4 Problem 1 Making a Tower Stable Together 1-2 The COSINE (COS) of an acute angle in a right triangle is the ratio of the length of the side that is adjacent to the angle to the length of the hypotenuse. Collaborate 3-6 (4 Minutes) cos 53=0.6 cos 37=0.8 cos 48=0.67 cos 42=0.74 cos 60=0.5 cos 30=0.87

9.4 Problem 1 Making a Tower Stable Together 7.

9.4 Problem 1 Making a Tower Stable Collaborate 8-12 (8 Minutes) Trigonometric Ratios sin 𝜃 = 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐿𝑒𝑔 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 cos 𝜃 = 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐿𝑒𝑔 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 tan 𝜃 = 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐿𝑒𝑔 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐿𝑒𝑔 SOH-CAH-TOA

9.4 Problem 1 Making a Tower Stable 8.

9.4 Problem 1 Making a Tower Stable

9.4 Problem 1 Making a Tower Stable 10.

9.4 Problem 1 Making a Tower Stable 11. 12.

9.4 Problem 1: Making a Tower Stable Together #13 tan 𝜃 = sin 𝜃 cos 𝜃 tan 𝜃 = 𝑂𝑝𝑝 𝐻𝑦𝑝 𝐴𝑑𝑗 𝐻𝑦𝑝 tan 𝜃 = 𝑂𝑝𝑝 𝐻𝑦𝑝 𝑥 𝐻𝑦𝑝 𝐴𝑑𝑗 tan 𝜃 = 𝑂𝑝𝑝 𝐴𝑑𝑗

9.4 Problem 2 Secant Ratio The secant of an angle is the reciprocal of the cosine of an angle. The SECANT (SEC) of an acute angle in a right triangle is the ratio of the length of the hypotenuse to the length of the side that is adjacent to the angle. We can always use the cosine function.

9.4 Problem 2 Secant Ratio On the calculator There is no secant button SEC is the reciprocal of COS sec 𝜃 = 1 cos 𝜃 Example 𝐹𝑖𝑛𝑑 sec 35 𝑜 = 1 cos 35 𝑜 ≈1.22

9.4 Problem 3 Inverse Cosine The inverse cosine (arc cosine or 𝑐𝑜𝑠 −1 ) is the measure of an acute angle whose cosine is x. The relationship of sides used is adjacent to hypotenuse. The calculator has an inverse cosine function. Only used to find the missing angle.

9.4 Problem 3 Inverse Cosine Collaborate 1-4 (4 Minutes)

9.4 Problem 3 Inverse Cosine

9.4 Problem 3 Inverse Cosine

9.4 Problem 3 Inverse Cosine

Example of SOH-CAH-TOA Find the measure of angle A using all 3 trig function Collaborate: 1 Minutes sin 𝐴 = 3 5 𝐴=𝑠𝑖𝑛 −1 3 5 ≈ 36.9 𝑜 5 cos 𝐴 = 4 5 𝐴=𝑐𝑜𝑠 −1 4 5 ≈ 36.9 𝑜 3 tan 𝐴 = 3 4 𝐴=𝑡𝑎𝑛 −1 3 4 ≈ 36.9 𝑜 A 4

Hypotenuse X Adjacent cos 𝜃 = 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 cos 65 = 𝑥 12 𝑥=12∙ cos 65 𝑥≈5.07 𝑓𝑒𝑒𝑡

Use the given diagram to find the missing side X 28 𝑜 100 ft cos 28 = 100 𝑥 𝑥 cos 28 =100 𝑥= 100 cos 28 ≈113.3 𝑓𝑒𝑒𝑡 Shortcut cos 28 = 100 𝑥 𝑆𝑤𝑖𝑡𝑐ℎ 𝑥= 100 𝑐𝑜𝑠 28 ≈113.3 𝑓𝑒𝑒𝑡

Use the given diagram to find the missing side 100 ft 28 𝑜 X cos 28 = 𝑥 100 100∗ cos 28 =𝑥 𝑥≈88.29 𝑓𝑒𝑒𝑡

Formative Assessment SOH-CAH-TOA Skills Practice 9.4 Problem Set Pg. 693-700 (1-43) odd cos 𝜃 = 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐿𝑒𝑔 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 sec 𝜃 = 1 cos 𝜃 SOH-CAH-TOA Check the MODE on the calculators The MODE must be in DEGREES