9.3 Trigonometry: Sine Ratio Geometry 9.3 Trigonometry: Sine Ratio

9.3 The Sine Ratio Sine Ratio, Cosecant Ratio, and Inverse Sine Objectives Use the sine ratio in a right triangle to solve for unknown side lengths. Use the cosecant ratio in a right triangle to solve for unknown side lengths. Relate the sine ratio to the cosecant ratio. Use the inverse sine in a right triangle to solve for unknown angle measures. SOH – CAH – TOA

9.3 Problem 1 Fore! Collaborate 1-3 (3 Minutes)

9.3 Problem 1 Fore! What affect does the angle of the club face have on the ball? Ratio of Opposite to Hypotenuse 21 𝑜 : 19.3 54 ≈0.36 35 𝑜 : 34.4 60 ≈0.57 39 𝑜 : 39 62 ≈0.63 What happens to the ratio as the angle gets larger?

9.3 Problem 1 Fore! The SINE (SIN) of an acute angle in a right triangle is the ratio of the length of the side that is opposite the angle to the length of the hypotenuse. Collaborate 4-10 (10 Minutes) Sine value refers to the sine of the angle measure. There are no missing pieces to solve for. We are “proving” that the ratio is equal to the sine of the angle. CALCULATOR MODE MUST BE IN DEGREES

9.3 Problem 1 Fore! #5 sin 21 = 19.3 54 ≈0.36 sin 69 = 50.4 54 ≈0.93

9.3 Problem 1 Fore! 9.

9.3 Problem 2 Cosecant Ratio The cosecant of an angle is the reciprocal of the sine of an angle. The COSECANT (CSC) 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 opposite the angle. We can always use the sine function.

9.3 Problem 2 Cosecant Ratio On the calculator There is no cosecant button CSC is the reciprocal of SIN csc 𝜃 = 1 sin 𝜃 Example 𝐹𝑖𝑛𝑑 c𝑠𝑐 35 𝑜 = 1 sin 35 𝑜 ≈1.74

9.3 Problem 3 Inverse Sine The inverse sine (arc sine or 𝑠𝑖𝑛 −1 ) is the measure of an acute angle whose sine is x. The relationship of sides used is opposite to hypotenuse. The calculator has an inverse sine function. Only used to find the missing angle.

9.3 Problem 3 Inverse Sine Collaborate 1-4 (5 Minutes)

9.3 Problem 3 Inverse Sine

9.3 Problem 3 Inverse Sine

9.3 Problem 3 Inverse Sine

9.3 Problem 3: Inverse Sine Together #5 sin 𝜃 = 𝑂𝑝𝑝 𝐻𝑦𝑝 𝜃= 𝑠𝑖𝑛 −1 3.9 56 𝜃≈ 3.99 𝑜 We have the opposite side Which side do we still need? The hypotenuse is, on average, 56 meters

𝜽 Hypotenuse Opposite sin 𝜃 = 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 sin 𝜃 = 4 14 𝜃= 𝑠𝑖𝑛 −1 4 14 𝜃≈ 16.6 𝑜

Use the given diagram to find the missing side X 100 ft 28 𝑜 sin 28 = 100 𝑥 𝑥 sin 28 =100 𝑥= 100 sin 28 ≈213 𝑓𝑒𝑒𝑡 Shortcut sin 28 = 100 𝑥 𝑆𝑤𝑖𝑡𝑐ℎ 𝑥= 100 𝑠𝑖𝑛 28 ≈213 𝑓𝑒𝑒𝑡

Use the given diagram to find the missing side 100 ft X 28 𝑜 sin 28 = 𝑥 100 100∗ sin 28 =𝑥 𝑥≈46.95 𝑓𝑒𝑒𝑡

Formative Assessment SOH-CAH-TOA Skills Practice 9.3 Problem Set Pg. 685-692 (1-43) Odd sin 𝜃 = 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐿𝑒𝑔 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 csc 𝜃 = 1 sin 𝜃 SOH-CAH-TOA Check the MODE on the calculators The MODE must be in DEGREES