1. 3.4 Finding an Angle otherwise known as Inverses!40 27 𝑋°
Reference angle is the unknown

2. Solving Trigonometric Equations
There are only three possibilities for the placement of the variable ‘x”.
Sin =
sin =
sin =
sin 25 =
sin x =
sin =
sin 25 =
sin 25 =
x =
x = (12) (sin 25)
x = 5.04 cm
x = 28.4 cm
Note you are looking for an angle here!

3. What does it mean to solve for an angle?
When we say sin 𝜃= we are saying sine of an angle 𝜃
is the value But what is 𝜃?
sin 𝜃 = 𝑂𝑝𝑝 𝐻𝑦𝑝
sin 𝜃= 12 25
𝜃= 𝑠𝑖𝑛 −1 𝑂𝑝𝑝 𝐻𝑦𝑝
How do we get 𝜃 by itself ?
Sometimes it’s not in fraction form
𝜃= 𝑠𝑖𝑛 − so
sin 𝜃 =𝑥
𝜃=28.69°
𝜃= 𝑠𝑖𝑛 −1 𝑥

4. Example:
sin x = find angle x.
𝑥= 𝑠𝑖𝑛 −1 (0.1115)
Hit the 2nd button
Then the sin button
This is what your screen should have
Hit enter!
Give 2 decimals and
don’t forget your degree
𝑥=6.40°

5. Try these Examples:
Solve for the following angles
3.𝑡𝑎𝑛𝜃= 3
2. cos 𝑥=0.8988
𝑥=26.00°
𝜃=60°
4. sin 𝜃= 1 2
5. cos 𝜃=
𝜃=30°
𝜃=38.50°
6. sin 𝑥=0.9997
7. tan 𝑥=2.7598
𝑥=88.60°
𝑥=70.08°

6. Step 1: What sides are we given in regards to the reference angle? K
Example 8:
Step 1: What sides are we given in regards to the
reference angle? K
Hyp
Opp
Adj 40 27
Step 2: Ask which trig function uses these sides? X° J G
si𝑛𝜃= 𝑜𝑝𝑝 ℎ𝑦𝑝
𝑐𝑜𝑠𝜃= 𝑎𝑑𝑗 ℎ𝑦𝑝
tan𝜃= 𝑜𝑝𝑝 𝑎𝑑𝑗
Step 3: Set Up equation using the known values
𝑥= 𝑠𝑖𝑛 −
sin⁡(𝑋°)= 27 40
𝑥=42.45°

7. Example 9:
Step 1: What sides are we given in regards to the
reference angle? P 𝜃
Hyp
Opp
Adj 11 6
Step 2: Ask which trig function uses these sides? L U
si𝑛𝜃= 𝑜𝑝𝑝 ℎ𝑦𝑝
𝑐𝑜𝑠𝜃= 𝑎𝑑𝑗 ℎ𝑦𝑝
tan𝜃= 𝑜𝑝𝑝 𝑎𝑑𝑗
Step 3: Set Up equation using the known values
𝑥= 𝑐𝑜𝑠 −
cos⁡(𝜃)= 6 11
𝜃=56.94°

8. Step 1: What sides are we given in regards to the reference angle?
Example 10:
Step 1: What sides are we given in regards to the
reference angle?
Hyp
Opp
Adj 17 𝜃
Step 2: Ask which trig function uses these sides? 13
si𝑛𝜃= 𝑜𝑝𝑝 ℎ𝑦𝑝
𝑐𝑜𝑠𝜃= 𝑎𝑑𝑗 ℎ𝑦𝑝
tan𝜃= 𝑜𝑝𝑝 𝑎𝑑𝑗
Step 3: Set Up equation using the known values
𝜃= 𝑡𝑎𝑛 −
tan⁡(𝜃)= 17 13
𝜃=52.59°

9. Now we can SOLVE for a triangle
Use your givens to find the missing parts
Remember you must have
at least
2 sides
Hypotenuse or
Leg
1 side and 1 angle 𝜃
Leg

10. 11. Solve for the triangle. B
∡𝐶= 20
𝐴𝐶 = A
30° C
𝐴𝐵 =

11. ∡𝐴= ∡𝑀= 𝑀𝑆 = 12. In ∆𝑆𝐴𝑀, ∡𝑆 is the right angle.
The length of side 𝐴𝑀 is 15, and the length
of side 𝑆𝐴 is 7. Find all angles and side
lengths in ∆𝑆𝐴𝑀.
∡𝐴=
∡𝑀=
Draw the triangle first!
𝑀𝑆 =

12. ∡𝐴= 𝐵𝐶 = 𝐴𝐵 = 13. In ∆𝐴𝐵𝐶, ∡𝐵 is the right angle.
The length of side 𝐴𝐶 is 10, and ∡𝐶=55°.
Find all angles and side lengths in ∆𝐴𝐵𝐶.
∡𝐴=
Draw the triangle first!
𝐵𝐶 =
𝐴𝐵 =