1. Section 5-5: Solving Right Triangles
Your calculator can find sine, cosine, and tangent in degrees and radians.
sin cos tan
We will use their definition to find secant, cosecant, and cotangent.
csc = 𝑠𝑖𝑛 sec = 𝑐𝑜𝑠 cot = 𝑡𝑎𝑛
When using the calculator, first you must determine if the angle is in DEGREES or RADIANS. Put your calculator in the appropriate MODE.
Example 1: Use the calculator to find the following.
a) cot 45° b) csc 30° c) sec 22.5°
INVERSE OF THE SINE, COSINE AND TANGENT FUNCTIONS
How do we solve : sin 𝜃 = -.5
1 tan 45°
1 sin 30°
1 cos 22.5°
= 1
= 2 =
𝜃 = sin −1 (−.5)
𝜃 = -30°

2. That means, sin-1 and sin will cancel each other out, as well as cos and cos-1, or tan and tan-1. We go backward on the calculator to find the angle in degrees or radians using the
sin called arcsine cos called arccosine tan called arctangent
Arcsine, arccosine and arctangent are ____________not functions.
When going backward, the best method for understanding your key sequence on the calculator is to solve algebraically for 𝜽 first, then type into the calculator.
Example 2: Find in degrees.
a) cos 𝜃 = b) sin 𝜃 = c) tan 𝜃 =
relations
𝜃= cos −1 (.5)
𝜃 = sin −1 (.7216)
𝜃 = tan −1 (1.1256)
𝜃 = 60°
𝜃 =
𝜃 =
𝜃 ≈ 46.2°
𝜃 ≈48.4°

3. Example 1:
If r = 14 and s = 8, find S.
You want to find the measure of an acute angle in a
right triangle. You know the side opposite the angle
and the hypotenuse.
sin S= 𝑠𝑖𝑑𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
sin S= 8 14
S= 𝑠𝑖𝑛 −1 4 7
Angle S is about 34.8°.
S =

4. Trigonometry can be used to find the angle of ______________ or the angle of
depression.
Example 2:
HOUSEHOLD The camera for a baby monitor is set up on a shelf in a child’s room, and it is angled so that it captures the image of the center of the baby’s crib The shelf is about 3 feet higher than the crib, and its horizontal distance from the crib is about 7 feet. What is the angle of depression of the light?
The angle of depression and the angle of elevation
are equal in measure because they are alternate
interior angles.
tan 𝜃= 𝑠𝑖𝑑𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
tan 𝜃= 3 7
𝜃= 𝑡𝑎𝑛 −
The angle of depression should be about 23.2°. 𝜃=

5. You can use trigonometric functions and inverse relations to solve right triangles. To solve a
right triangle means to find __________________________ and ____________.
all the measures of its sides
the angles
*** Whenever possible, use the measures given in the problem to find the unknown measures.
Example 3:
Solve each triangle described, given the triangle at the right.
A. B = 42°, b = 4.5
Find A : A = 180
A = 48°
Find a : tan 48°= 𝑎 4.5
Find c : cos 48°= 4.5 𝑐
4.5 tan 48°=𝑎
𝑐 cos 48°=4.5
a =
𝑐= cos 48°
𝑎 ≈5
c =
𝑐 ≈6.7

6. B b = 18, c = 52
Find a : a2 + (18)2 = (52)2
a = 2704
a2 = 2380
a =
𝑎 ≈
𝑎 ≈48.8
Find A : sin 𝐴=
A = 𝑠𝑖𝑛 −
A =
A = 69.8°
Find B : °+𝐵+90°=180°
B = 20.2°