1. Introduction
In the real world, if you needed to verify the size of a television, you could get some measuring tools and hold them up to the television to determine that the TV was advertised at the correct size. Imagine, however, that you are a fact checker for The Guinness Book of World Records. It is your job to verify that the tallest building in the world is in fact Burj Khalifa, located in Dubai. Could you use measuring tools to determine the size of a building so large? It would be extremely difficult and impractical to do so.
2.2.1: Calculating Sine, Cosine, and Tangent

2. Introduction, continued
You can use measuring tools for direct measurements of distance, but you can use trigonometry to find indirect measurements. First, though, you must be able to calculate the three basic trigonometric functions that will be applied to finding those larger distances. Specifically, we are going study and practice calculating the sine, cosine, and tangent functions of right triangles as preparation for measuring indirect distances.
2.2.1: Calculating Sine, Cosine, and Tangent

3. Key Concepts
The three basic trigonometric ratios are ratios of the side lengths of a right triangle with respect to one of its acute angles.
As you learned previously:
Given the angle
2.2.1: Calculating Sine, Cosine, and Tangent

4. Key Concepts, continued
Notice that the trigonometric ratios contain three unknowns: the angle measure and two side lengths.
Given an acute angle of a right triangle and the measure of one of its side lengths, use sine, cosine, or tangent to find another side.
Use the inverses of these trigonometric functions
(sin–1, cos–1, and tan–1) to find the acute angle measures given two sides of the right triangle.
2.2.1: Calculating Sine, Cosine, and Tangent

5. Common Errors/Misconceptions
not using the given angle as the guide in determining which side is opposite or adjacent
forgetting to ensure the calculator is in degree mode before completing the operations
using the trigonometric function instead of the inverse trigonometric function when calculating the acute angle measure
2.2.1: Calculating Sine, Cosine, and Tangent

6. Guided Practice 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.
2.2.1: Calculating Sine, Cosine, and Tangent

7. Guided Practice
2.2.1: Calculating Sine, Cosine, and Tangent

8. Guided Practice: Example 2, continued
Determine which trigonometric function to use by identifying the given information.
Given an angle of w°, 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.
2.2.1: Calculating Sine, Cosine, and Tangent

9. Guided Practice: Example 2, continued
Cosine is the trigonometric function that uses
adjacent and hypotenuse,
so we will use it to calculate the angle the truck drove to reach the bottom of the road.
2.2.1: Calculating Sine, Cosine, and Tangent

10. Guided Practice: Example 2, continued
Set up an equation using the cosine function and the given measurements.
Therefore,
Solve for w.
2.2.1: Calculating Sine, Cosine, and Tangent

11. Guided Practice: Example 2, continued
Solve for w by using the inverse cosine since we are finding an angle instead of a side length.
2.2.1: Calculating Sine, Cosine, and Tangent

12. Guided Practice: Example 2, continued
Use a calculator to calculate the value of w.
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. Press [ENTER].
Step 3: Arrow right to DEGREE.
Step 4: Press [ENTER]. The word “DEGREE” should be highlighted inside a black rectangle.
2.2.1: Calculating Sine, Cosine, and Tangent

13. Guided Practice: Example 2, continued
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: Press [2ND][COS][990][÷][1027][)].
Step 2: Press [ENTER].
w = , or 15°.
2.2.1: Calculating Sine, Cosine, and Tangent

14. Guided Practice: Example 2, continued
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].
2.2.1: Calculating Sine, Cosine, and Tangent

15. ✔ Guided Practice: Example 2, continued Next, perform the calculation.
Step 1: In the calculate window from the home screen, press [cos–1][990][÷][1027].
Step 2: Press [enter].
w = , or 15°
The trucker drove at an angle of 15°
to the top of the hill. ✔
2.2.1: Calculating Sine, Cosine, and Tangent

16. Guided Practice: Example 2, continued
2.2.1: Calculating Sine, Cosine, and Tangent

17. Guided Practice Example 4 Solve the right triangle.
Round sides to the nearest
thousandth.
2.2.1: Calculating Sine, Cosine, and Tangent

18. Guided Practice: Example 4, continued Find the measures of and .
Solving the right triangle means to find all the missing angle measures and all the missing side lengths. The given angle is 64.5° 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
2.2.1: Calculating Sine, Cosine, and Tangent

19. Guided Practice: Example 4, continued Recall that
2.2.1: Calculating Sine, Cosine, and Tangent

20. Guided Practice: Example 4, continued
On a TI-83/84:
Step 1: Press [17][TAN][64.5][)].
Step 2: Press [ENTER].
x =
On a TI-Nspire:
Step 1: In the calculate window from the home screen, press [17][tan][64.5].
Step 2: Press [enter].
The measure of AC =
2.2.1: Calculating Sine, Cosine, and Tangent

21. Guided Practice: Example 4, continued
To find the measure of , either acute angle may be used as a reference. 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.
2.2.1: Calculating Sine, Cosine, and Tangent

22. Guided Practice: Example 4, continued
Use the cosine function based on the given information.
Recall that
2.2.1: Calculating Sine, Cosine, and Tangent

23. Guided Practice: Example 4, continued
2.2.1: Calculating Sine, Cosine, and Tangent

24. Guided Practice: Example 4, continued
On a TI-83/84:
Check to make sure your calculator is in DEGREE mode first. Refer to the directions in the previous example.
Step 1: Press [17][÷][COS][64.5][)].
Step 2: Press [ENTER].
y =
2.2.1: Calculating Sine, Cosine, and Tangent

25. Guided Practice: Example 4, continued
On a TI-Nspire:
Check to make sure your calculator is in degree mode first. Refer to the directions in the previous example.
Step 1: Press [trig][17][÷][cos][64.5].
Step 2: Press [enter].
y =
The measure of AB =
2.2.1: Calculating Sine, Cosine, and Tangent

26. Guided Practice: Example 4, continued
Use the Pythagorean Theorem to check your trigonometry calculations.
AC = and AB =
The answer checks out.
2.2.1: Calculating Sine, Cosine, and Tangent

27. Guided Practice: Example 4, continued Find the value of ∠A.
2.2.1: Calculating Sine, Cosine, and Tangent

28. Guided Practice: Example 4, continued
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.
2.2.1: Calculating Sine, Cosine, and Tangent

29. Guided Practice: Example 4, continued
Use the inverse trigonometric function since you are solving for an angle measure.
2.2.1: Calculating Sine, Cosine, and Tangent

30. Guided Practice: Example 4, continued
On a TI-83/84:
Step 1: Press [2ND][SIN][17][÷][39.488][)].
Step 2: Press [ENTER].
z = °
On a TI-Nspire:
Step 1: Press [trig][sin][17][÷][39.488].
Step 2: Press [enter].
2.2.1: Calculating Sine, Cosine, and Tangent

31. ✔ Guided Practice: Example 4, continued
Check your angle measure by using the Triangle Sum Theorem.
∠A is 25.5°
The answer checks out. ✔
2.2.1: Calculating Sine, Cosine, and Tangent

32. Guided Practice: Example 4, continued
2.2.1: Calculating Sine, Cosine, and Tangent