8.4 Applying Trigonometric Ratios Wednesday, July 18, 2018 Warm Up Write each fraction as a decimal. 1. 2 3 = 2. Sin(23) = Solve each equation. 0.8= 5.8 𝑥 4. 0.94= 𝑥 8.5 Let me know if you want to be a student aide 0.67 0.3907 x = 7.25 x = 7.99

Read Only The hypotenuse is always the longest side of a right triangle. So the denominator of a sine or cosine ratio is always greater than the numerator. Therefore the sine and cosine of an acute angle are always positive numbers less than 1. Since the tangent of an acute angle is the ratio of the lengths of the legs, it can have any value greater than 0.

The Pilatusbahn

Example 1 The Pilatusbahn in Switzerland is the world’s steepest cog railway. Its steepest section makes an angle of about 26 with the horizontal and rises about 0.9 km. To the nearest ten-thouosandth of a kilometer, how long is this section of the railway track?

Step One: Make a sketch. B 0.9 km 26 C A

Step Two: Organize the given information. θ = 26° Opposite (θ) = 0.9 km Hypotenuse = X The two sides we care about are the opposite and hypotenuse. Therefore the trig function we want is sin.

Step Three: Set up your equation. 𝑆𝑖𝑛 θ = 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 Write a trigonometric ratio. 𝑆𝑖𝑛 26 = 0.9 𝑥 Substitute the given values.

Step Four: Solve your equation. 0.4384= 0.9 𝑥 Find Sin(26) Cross multiply 0.4384𝑥=0.9 .04384𝑥 0.4384 = 0.9 0.4384 Divide by coefficient Simplify the expression. *Round to ten-thousandth 𝑥=2.0529 𝑘𝑚

What is “SOH-CAH-TOA”? 𝑆𝑖𝑛 θ = 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝐶𝑜𝑠 θ = 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑇𝑎𝑛 θ = 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡

If you are given a ratio how do you find the angle value? This is the opposite of using the angle value to find the ratio of the side. Step 1: Go to the given column (Sine, Cosine, or tangent). Step 2: Scan that column to find the decimal closest to your ratio. Step 3: Find the angle value for that row. This is your angle value.

Example 2 Sin(θ) = .22 Cos(θ) = .22 Tan(θ) = .22

The Stratosphere

Example 3 The Stratosphere is 1,149 ft tall. Batman zip lines from the top and glides at a 10° angle from the horizon. If he travels a horizontal distance of 6,516.3 ft, how long must his zip line be.

Step One: Make a sketch. B 10 x C A 6,516.3 ft

Step Two: Organize the given information. θ = 10° Adjacent (θ) = 6516.3 Hypotenuse = x ft The two sides we care about are the adjacent and hypotenuse. Therefore the trig function we want is cosine.

Step Three: Set up your equation. 𝐶𝑜𝑠 θ = 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 Write a trigonometric ratio. 𝐶𝑜𝑠 10 = 6,516.3 𝑥 Substitute the given values.

Step Four: Solve your equation. 𝐶𝑜𝑠 10 = 6,516.3 𝑥 Find Cos(10) Cross multiply 0.9848𝑥=6,516.3 0.9848𝑥 0.9848 = 6,516.3 0.9848 Divide by coefficient Simplify the expression. *Round to ten-thousandth 𝑥=6,616.8765 𝑓𝑡

The Hyperion Tree

Example 1 The Hyperion tree is the world’s tallest tree. You stand at the base of the tree and walk half a mile away. You then measure the angle to the top of the tree to be 8.2°. How tall is the tree.

Step One: Make a sketch. B x 8.2 C A 0.5 miles

Step Two: Organize the given information. θ = 8.2° Opposite (θ) = x Adjacent = 2640ft The two sides we care about are the opposite and adjacent. Therefore the trig function we want is tangent.

Step Three: Set up your equation. 𝑇𝑎𝑛 θ = 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 Write a trigonometric ratio. 𝑇𝑎𝑛 8.2 = 𝑥 2640 Substitute the given values.

Step Four: Solve your equation. 0.1405= 𝑥 2640 Find Tan(8) Cross multiply 𝑥=0.1405∙2640 𝑥=370.92 Simplify