1. 9.6 Solving Right Triangles
Unit IIC Day 6

2. Do Now
Fill in the ratios with words.
sin(θ) =
cos(θ) =
tan(θ) =

3. Solving a right triangle
Every right triangle has one right angle and two _________ angles.
Every right triangle has one _____________ and two __________.
To solve a right triangle means to determine the measures of all ______ parts. You can solve a right triangle if given one of the following situations:
Two side lengths
One side length and one acute angle measure
As you learned in Lesson 9.5, you can use the side lengths of a right triangle to find trigonometric ratios for the acute angles of the triangle. As you will see in this lesson, once you know the sine, cosine, or tangent of an acute angle, you can use a calculator to find the measure of the angle.

4. Inverse Functions
The inverse of a function switches its inputs and outputs.
Example: For f(x) = x2, if you input 3, the output is 9. Its inverse is f(x) = √x: If you input 9, the output is 3.
tan(θ) takes ___________ as its input and gives ________________ as its output,
Its inverse, tan– 1(x), takes _______________ as its input and gives _____________ as its output.
Example: sin(30º) = ___, so sin-1(___) = ____
an angle; the ratio opp/adj
the ratio opp/adj; an angle

5. Example 1:
Solve the right triangle. Round the decimals to the nearest tenth.
Start by using the Pythagorean Theorem. You have side a and side b. You don’t have the hypotenuse which is side c—directly across from the right angle.
c = √13 ≈ 3.6
Then use a calculator to find the measure of B: tan-1 (2/3) ≈ 33.7°
Because A and B are complements, you can write mA = 90° - mB ≈ 90° – 33.7° = 56.3°

6. Ex. 2: Solving a Right Triangle
Solve the right triangle. Round decimals to the nearest tenth.
For h, you are looking for opposite and hypotenuse which is the sin ratio.
sin(H) = opp/ hyp
sin(25) = h/13
13(0.4226) ≈ h
5.5 ≈ h
For g, you are looking for adjacent and hypotenuse which is the cosine ratio.
cos(H) = adj/ hyp
cos(25) = g/13
13(0.9063) ≈ g
11.8 ≈ g

7. Ex: 3: Real Life
When the shuttle’s altitude is about miles, its horizontal distance to the runway is about 59 miles. What is its glide angle? Round your answer to the nearest tenth.
Space Shuttle: During its approach to Earth, the space shuttle’s glide angle changes.
You know opposite and adjacent sides. If you take the opposite and divide it by the adjacent sides, then take the inverse tangent of the ratio, this will yield you the slide angle.
tan(x) = opp/adj
tan(x) = 15.7/59
tan-1 (15.7/59) ≈ 14.9°

8. Ex: 3: Real Life
When the space shuttle is 5 miles from the runway, its glide angle is about 19°. Find the shuttle’s altitude at this point in its descent. Round your answer to the nearest tenth.
tan(19) = opp/adj
tan(19) = h/5
5tan(19) = h
1.7 miles ≈ h

9. Closure
What is the minimum amount of information you need to solve a right triangle?
2 side lengths or 1 side length and 1 acute angle measure.