1. 8-4 Trigonometry The student will be able to:
Find trigonometric ratios using right triangles.
Use trigonometric ratios to find angle measures in right triangles.

2. Trigonometric Ratios
Trigonometric ratio – a ratio of the lengths of two sides of a right triangle.
By AA Similarity, a right triangle with the same acute angle measure is similar to every other right triangle with the same acute angle measure.
In a right triangle, if you know the measures of two sides or if you know the measures of one side and an acute angle, then you can find all the measures of the missing sides or angles of the triangle.
The three most common trigonometric ratios are:

3. Example 1: Find sin J, cos J, tan J, sin K, cos K, and tan K
Example 1: Find sin J, cos J, tan J, sin K, cos K, and tan K. Express each ratio as a fraction and as a decimal to the nearest hundredth.
≈ 0.38
≈ 0.92
≈ 0.42
≈ 0.92
≈ 0.38
≈ 2.4

4. Example 2: Use a special right triangle to express the cosine of 45° as a fraction and as a decimal to the nearest hundredth.
1st – Draw a 45°-45°-90° right triangle, label the side lengths with x as the length of the legs. x
2nd – How do we find cos? x
As a fraction.
As a decimal.

5. Example 3: Real Life Situation The front of the vacation cottage shown is an isosceles triangle. What is the height (x) of the cottage above its foundation? What is the length (y) of the roof? Explain your reasoning.
Hint: If it’s an isosceles triangle, then the base is bisected.

6. 2nd - How can you find the measure of x with the 60° angle?
1st – This is a 30°-60°-90° right triangle. Which acute angle do we know?
60°
Hint: When you are given the angle degree your calculator must be in degree mode.
32.5
2nd - How can you find the measure of x with the 60° angle?
Adjacent side
Opposite side
3rd – Which trigonometric ratio uses the opposite side & the adjacent side?
4th – What is the length of the roof?
65 ft
It’s an equilateral triangle.
(32.5)tan 60 = x
All sides are equal.
56 ft ≈ x

7. You Try It: 1. Express each ratio of angle L as a fraction and as a decimal to the nearest hundredth A fitness trainer sets the incline on a treadmill to 7°. The walking surface is 5 feet long. Approximately how many inches did the trainer raise the end of the treadmill from the floor.
≈ 0.32
≈ 0.95
≈ 0.34
≈ 7.3 in
y = (60)sin 7
y ≈ 7.3 in

8. Use Inverse Trigonometric Ratios
To find an angle measure when you have two side measures:
1. Determine which trig ratio applies.
2. Use the inverse of the trig function on your calculator (Hint: use the 2nd key).
3. The total of the angle measures may not equal 180° due to rounding.
4. Round angles to the nearest degree and sides to the nearest tenth.
If sin N = x, then sin-1 x =
If cos N = x, then cos-1 x =
If tan N = x, then tan-1 x =

9. Example 4: Find x. Round to the nearest degree if necessary.
1st – Which angle are you looking for?
2nd – Which trig ratio applies?
3rd – What’s the inverse?

10. Example 5: Use a calculator to find the measure of to the nearest degree if necessary.
1st – Which angle are you looking for?
2nd – Which trig ratio applies?
3rd – What’s the inverse?

11. Solving a Right Triangle
Remember: To solve a right triangle you must know (1) two side lengths or (2) one side length and the measure of one acute angle.
Solve the right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree.
1st – You’re given two sides. Find the measure of the missing side. 12
a2 + b2 = c2
b2 = 144
52 + b2 = 132
b = 12
25 + b2 = 169
3rd – Solve for the remaining acute angle.
2nd – Choose one of the acute angles and solve for it. Let’s start with F.

12. Example 6: Solve the right triangle
Example 6: Solve the right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree.
4.7
1st – You’re given one side and one angle. Find the measure of one of the missing sides.
8.8
(10)cos 62 = x
4.7 ≈ x
2nd – You know two sides. Find the measure of the missing side.
3rd – Find the measure of the missing acute angle.
a2 + b2 = c2
b2 = 77.9
b2 = 102
b ≈ 8.8
b2 = 100

13. You Try it: 1. Use a calculator to find the measure of to the nearest degree.47
2. Solve the right triangle. Round side measures to the nearest hundredth and angle measures to the nearest degree.
a2 + b2 = c2
= c2
= c2
65 = c2
8.06 = c