This document discusses solving trigonometric equations. It provides examples of solving trigonometric equations by using trigonometric identities to isolate the variable. Solutions may involve factoring expressions or using trigonometric properties like csc^2(x) = 1 + cot^2(x). The document also notes that some equations have no solution if the trigonometric function values are outside their normal range. Students are given examples to practice solving trigonometric equations.

1. Solving
Trigonometric
Equations

2. Objectives
o Solving Trigonometric Equations.
o Find extraneous solutions from trigonometric equations.

3. Vocabulary
 Trigonometric Equations.

4. Basic
trigonometry
identities

5. Remember that solving a trigonometric equations means solving for all values of the variable.
Solve 2 𝑠𝑒𝑐2 ∅ − 𝑡𝑎𝑛4 ∅ = −1,for all values of ∅ if ∅ is measured by degrees.
Solution:
*use 𝑠𝑒𝑐2 ∅ = 1 + 𝑡𝑎𝑛2 ∅
2 1 + 𝑡𝑎𝑛2
∅ − 𝑡𝑎𝑛4
∅ = −1
* Use distributive property:
2 + 2𝑡𝑎𝑛2
∅ − 𝑡𝑎𝑛4
∅ = −1
*Set one side of the equation equal to 0
𝑡𝑎𝑛4 ∅ − 2𝑡𝑎𝑛2 ∅ − 3 = 0
*Factor
𝑡𝑎𝑛2
∅ − 3 𝑡𝑎𝑛2
∅ + 1 = 0
*Use product property:
𝑡𝑎𝑛2∅ − 3 =0 or ( 𝑡𝑎𝑛2∅ + 1) = 0
𝑡𝑎𝑛2
∅ = 3 𝑡𝑎𝑛2
∅ = −1 (𝑡ℎ𝑖𝑠 𝑝𝑎𝑟𝑡 𝑔𝑖𝑣𝑒𝑠 𝑛𝑜 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑠𝑖𝑛𝑐𝑒 𝑡𝑎𝑛2
∅ 𝑖𝑠 𝑛𝑒𝑣𝑒𝑟 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 )
𝑡𝑎𝑛2∅ = ∓ 3
∅ = 60° + 180°𝑘 and ∅ = −60° + 180°𝑘, where 𝑘 is any integer
Solving Trigonometric Equations by using Identities

6. Solve the equation sin ∅ cot ∅ − cos2∅ = 0
Solution:
*cot ∅ =
cos∅
sin∅
sin ∅
cos∅
sin∅
− cos2∅ = 0
cos ∅ − cos2
∅ = 0
*Factor cos ∅
cos ∅ (1 − cos ∅) = 0
*Use product property:
cos ∅ = 0 or 1 − cos ∅ = 0
∅ =
π
2
+ 2k or cos ∅ = 1
Then, cos ∅ = 1
cos−1
1 = ∅
∅ = 0
Solving Trigonometric Equations by using Identities

7. Solving
Trigonometric
Equations by using
Identities
practice
 Solve 9 + 𝑐𝑜𝑡2 𝑥 = 12 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑥.

8. extraneous
solutions
 Some trigonometric equations have no solution. For
example, the equation 𝑐𝑜𝑠∅ = 4 has no solution
because all values of 𝑐𝑜𝑠∅ are between −1 𝑎𝑛𝑑 1.
Thus, the solution of 𝑐𝑜𝑠∅ = 4 is empty.

9. When you ride a Ferris wheel that has a diameter of 40 meters and turn at a rate of 1.5 revolutions per minute, the height
above the ground in meters of your seat after 𝑡 minutes can be by the equation ℎ = 21 − 20 cos 3𝜋𝑡.
After the ride begins, how long is it before your seat is 31 meters above the ground for the first time?
Solution:
*Replace ℎ with 31.
31 = 21 − 20 cos 3𝜋𝑡
*subtract 21 from each sides
10 = −20 cos 3𝜋𝑡
*take the arccosine
cos−1 −
1
2
= 3𝜋𝑡
The arccosine of −
1
2
𝑖𝑠
2𝜋
3
𝑜𝑟
4𝜋
3
So,
2𝜋
3
= 3𝜋𝑡 or
4𝜋
3
= 3𝜋𝑡
2𝜋
3
+ 2𝜋𝑘 = 3𝜋𝑡 or
4𝜋
3
+ 2𝜋𝑘 = 3𝜋𝑡
* Divide each term by 3𝜋
2
9
+
2
3
𝑘 = 𝑡 or
4
9
+
2
3
𝑘 = 𝑡
The least positive value for 𝑡 is obtained by letting 𝑘 = 0 in the first expression.
Therefore, 𝑡 =
2
9
of a minute or 13 seconds.
Solve trigonometric equations

10. Solve
trigonometric
equations
practice
 The tallest building in the world Burj Khalifa in Dubai, UAE,
with 2,717 feet.
What is the measure of ∅ if the length of its shadow is
1 mile?