6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios
Instruction
2. Set up an equation using the sine function and the given measurements.
V = 34º
opposite side = x hypotenuse = 150 ft Therefore, sin34º
"150x .
3. Solve for x by multiplying both sides of the equation by 150.
150 • sin 34º = x
4. Use a calculator to determine the value of x.
On a TI-83/84:
First, make sure your calculator is in Degree mode.
Step 1: Press [MODE].
Step 2: Arrow down twice to RADIAN.
Step 3: Arrow right to DEGREE.
Step 4: Press [ENTER]. The word “DEGREE” should be highlighted inside a black rectangle.
Step 5: Press [2ND].
Step 6: Press [MODE] to QUIT.
Note: You will not have to change to Degree mode again unless you have changed your calculator to Radian mode.
Next, perform the calculation.
Step 1: Enter [150][w ][SIN][34][)].
Step 2: Press [ENTER].
x = 83.879 (continued)
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios
Instruction
On a TI-Nspire:
First, make sure the calculator is in Degree mode.
Step 1: Choose 5: Settings & Status, then 2: Settings, and 2: Graphs and Geometry.
Step 2: Move to the Geometry Angle field and choose “Degree”.
Step 3: Press [tab] to “ok” and press [enter].
Then, set the Scratchpad in Degree mode.
Step 1: In the calculate window from the home screen, press [doc].
Step 2: Select 7: Settings and Status, then 2: Settings, and 1:
General.
Step 3: Move to the Angle field and choose “Degree”.
Step 4: Press [tab] to “Make Default” and press [enter] twice to apply this as the new default setting.
Next, perform the calculation.
Step 1: In the calculate window from the home screen, enter (150), then press [w][trig]. Use the keypad to select “sin,” then type 34.
Step 2: Press [enter].
x = 83.879
The length of Leo’s courtyard is about 84 feet.
Example 2
A trucker drives 1,027 feet up a hill that has a constant slope. When the trucker reaches the top of the hill, he has traveled a horizontal distance of 990 feet. At what angle did the trucker drive to reach the top? Round your answer to the nearest degree.
990 ft
w˚ 1027 ft
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios
Instruction
1. Determine which trigonometric function to use by identifying the given information.
Given an angle of wº, the horizontal distance, 990 feet, is adjacent to the angle.
The distance traveled by the trucker is the hypotenuse since it is opposite the right angle of the triangle.
adjacent to w˚
w˚ hypotenuse
Cosine is the trigonometric function that uses adjacent and hypotenuse, cosθ = adjacent
hypotenuse, so we will use it to calculate the angle the truck drove to reach the bottom of the road.
Set up an equation using the cosine function and the given measurements.
V = wº
adjacent leg = 990 ft hypotenuse = 1027 ft Therefore, cos w" 990
1027. Solve for w.
Solve for w by using the inverse cosine since we are finding an angle instead of a side length.
cos−
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios
Instruction
2. Use a calculator to calculate the value of w.
On a TI-83/84:
Check to make sure your calculator is in Degree mode first. Refer to the directions in Example 1.
Step 1: Press [2ND][COS][990][{ ][1027][)].
Step 2: Press [ENTER].
w = 15.426, or 15º.
On a TI-Nspire:
Check to make sure your calculator is in Degree mode first. Refer to the directions in Example 1.
Step 1: In the calculate window from the home screen, press [trig]
to bring up the menu of trigonometric functions. Use the keypad to select “cos–1.” Enter 990, then press [{] and enter 1027.
Step 2: Press [enter].
w = 15.426, or 15º.
The trucker drove at an angle of 15º to the top of the hill.
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios
Instruction
Example 3
In -TRY , Y is a right angle and tan T = 8
15. What is sin R ? Express the answer as a fraction and as a decimal.
1. Draw a diagram.
15 x 8
θ
R
Y T
2. Use the Pythagorean Theorem to find the hypotenuse.
82 + 152 = x2 64 + 225 = x2
"
289 x x = 17
3. Calculate sin R.
sinR .
= opposite = x = ≈ hypotenuse
15 15
17 0 882
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios
Instruction
Example 4
Solve the right triangle below. Round sides to the nearest thousandth.
17
B
C
A
64.5˚
1. Find the measures of AC and AB .
Solving the right triangle means to find all the missing angle measures and all the missing side lengths. The given angle is 64.5º and 17 is the length of the adjacent side. With this information, we could either use cosine or tangent since both functions’ ratios include the adjacent side of a right triangle. Start by using the tangent function to find AC . Recall that tanθ =opposite
adjacent .
17
B
C
A
64.5˚
x
"
tan 64.5˚
17 x
17 • tan 64.5º = x
(continued)
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios
Instruction
On a TI-83/84:
Step 1: Press [17][TAN][64.5][)].
Step 2: Press [ENTER].
x = 35.641 On a TI-Nspire:
Step 1: In the calculate window from the home screen, enter 17, then press [trig] to bring up the menu of trigonometric functions. Use the keypad to select “tan,” then enter 64.5.
Step 2: Press [enter].
x = 35.641
The measure of AC = 35.641.
To find the measure of AB , either acute angle may be used as an angle of interest. Since two side lengths are known, the Pythagorean Theorem may be used as well.
Note: It is more precise to use the given values instead of approximated values.
17
B
C
A
64.5˚
y
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios
Instruction
2. Use the cosine function based on the given information.
Recall that cosθ = adjacent hypotenuse. θ = 64 5. º
adjacent leg = 17 hypotenuse = y
Check to make sure your calculator is in Degree mode first. Refer to the directions in Example 1.
Step 1: Press [17][{ ][COS][64.5][)].
Step 2: Press [ENTER].
y = 39.488 On a TI-Nspire:
Check to make sure your calculator is in Degree mode first. Refer to the directions in Example 1.
Step 1: In the calculate window from the home screen, enter 17, then press [{ ][trig]. Use the keypad to select “cos,” and then enter 64.5.
Step 2: Press [enter].
y = 39.488
The measure of AB = 39.488.
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios
Instruction
3. Use the Pythagorean Theorem to check your trigonometry calculations.
172 + 35.6412 = y2 289 + 1267.36 = y2 1559.281 = y2
1559 281. = y
y = 39.488 The answer checks out.
AC = 35.641 and AB = 39.488.
4. Find the value of A .
17
B
C
A
64.5˚
39.488
z˚ 35.641
Using trigonometry, you could choose any of the three functions since you have solved for all three side lengths. In an attempt to be as precise as possible, let’s choose the given side length and one of the approximate side lengths.
sin
z" 17. 39 488
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios
Instruction
5. Use the inverse trigonometric function since you are solving for an angle measure.
arcsin 17 39.488
z=
On a TI-83/84:
Step 1: Press [2ND][SIN][17][{ ][39.488][)].
Step 2: Press [ENTER].
z = 25.500º On a TI-Nspire:
Step 1: In the calculate window from the home screen, press [trig]
to bring up the menu of trigonometric functions. Use the keypad to select “sin–1,” and then enter 17, press [{ ], and enter 39.488.
Step 2: Press [enter].
z = 25.500º
Check your angle measure by using the Triangle Sum Theorem.
m∠ +A 64 5. +90=180 m∠ +A 154 5. =180
m∠ = 25 5A . The answer checks out.
A is 25.5º.
6/*5t RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios
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