1. Class Greeting

2. Chapter 8 – Lesson 4
Trigonometric Ratios

3. Objective
Use trigonometric ratios to find angle measures in right triangles and to solve real-world problems.

4. Using given measures to find the unknown angle measures or side lengths of a triangle is known as solving a triangle.
To solve a right triangle, you need to know two side lengths or one side length and an acute angle measure.
We can use trigonometric ratios to solve right triangles.

5. Basic Trigonometric Ratios
SOH-CAH-TOA

6. Example 1: Trigonometric Ratios
Find the Trigonometric Ratios of each angle.

7. Find sin A, cos A, tan A, sin B, cos B, and tan B.
Example 4-1f

8. Answer:
Example 4-1g

9. Hypotenuse = 2(shorter leg)
Remember this from 8.3?
Find the values of x and y. Give your answers in simplest radical form.
Hypotenuse = 2(shorter leg)
22 = 2x
11 = x
Now we can find the sin 30° by using our trigonometric functions.
Now use your calculator to find the sin 30°.
KEYSTROKES:
SIN
ENTER
Therefore the sin 30° = 0.5.

10. Since the sin 30° = 0.5.
Conversely… if we know that the sin  1 = 0.5,
what is the measure of  1?
m  1 = 30°
This is written as sin-1(0.5) = 30°. (Try it on your calculator)
Finds the ratio of the sides.
Finds the measure of the angle.

11. Example 2: Calculating Angle Measures from Trigonometric Ratios
Use your calculator to find each angle measure to the nearest degree.
A. cos-1(0.87)
B. sin-1(0.85)
C. tan-1(0.71)
cos-1(0.87)  30°
sin-1(0.85)  58°
tan-1(0.71)  35°

12. Now we can use our calculator to find the angle measures.
Remember this?
Find sin A, cos A, tan A, sin B, cos B, and tan B.
Now we can use our calculator to find the angle measures.
Example 4-1f

13. Do these answers make sense? m B  53°
Remember your answers?
Answer:
Now use your calculator to find the angle measures.
m A
 37°
Do these answers make sense?
m B
 53°
Example 4-1g

14. Do these answers make sense?
Find sin A, cos A, tan A, sin B, cos B, and tan B.
 53°
 37°
The answers make sense, A and B are complementary.
Example 4-1f

15. Example 3: Solving a Right in ∆ the Coordinate Plane
The coordinates of the vertices of ∆PQR are P(–3, 3), Q(2, 3), and R(–3, –4). Find the side lengths to the nearest hundredth and the angle measures to the nearest degree.
Step 1 Find the side lengths. Plot points P, Q, and R.
PR = PQ = 5 Y P Q X
mP = 90° R
mR  90° – 54°  36°

16. Find the unknown measures
Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree.
c2 = a2 + b2
(5.7)2 = 52 + ST2
Since the acute angles of a right triangle are complementary, mT  90° – 29°  61°.

17. Example 4
Baldwin St. in Dunedin, New Zealand, is the steepest street in the world. It has a grade of 38%. To the nearest degree, what angle does Baldwin St. make with a horizontal line?
A 38% grade means the road rises (or falls) 38 ft for every 100 ft of horizontal distance.
100 ft
38 ft A B C

18. Use a calculator to find tan to the nearest ten thousandth.
Review
Use a calculator to find tan to the nearest ten thousandth.
KEYSTROKES:
TAN
ENTER
Answer:
Example 4-2a

19. Kahoot!

20. Lesson Summary: Objective
Use trigonometric ratios to find angle measures in right triangles and to solve real-world problems.

21. Preview of the Next Lesson:
Objective
Solve problems involving angles of elevation and angles of depression.

22. Stand Up Please

23. Lesson Quiz: Part I
Use your calculator to find each angle measure to the nearest degree.
1. cos-1 (0.97)
2. tan-1 (2)
3. sin-1 (0.59)

24. Lesson Quiz: Part II
Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree.

25. Lesson Quiz: Part III
6. The coordinates of the vertices of ∆MNP are M (–3, –2), N(–3, 5), and P(6, 5). Find the side lengths to the nearest hundredth and the angle measures to the nearest degree.