Solving Trigonometric Equations: Tips and Examples, Exercises of Mathematics

Download Solving Trigonometric Equations: Tips and Examples and more Exercises Mathematics in PDF only on Docsity!

Pre-Calculus

Quarter 2 – Module 8 :

Trigonometric Equations

Lesson

1

Solving Trigonometric

Equation

Any equation that involves trigonometric expressions is called a trigonometric equation. To “solve an equation” means to find all the solutions of the equation. In this module you will be learning how to solve trigonometric equations and solve situational problems that involves trigonometric equations.

What’s In

Recall the table of trigonometric values of special angles in the unit circle, trigonometric identities and formulas. These will help you answers problems involving trigonometric equations.

Angles in Radian

0

Angles in Degree

0 3 0 4 5 6 0 9 0 120 135 150 180 210 225 240 270 300 315 330 360

Sin 0 √

√ 1 √ √

0 √

√ 1 √ √

0

Cos (^1) √ √

0 √

√ 1 √ √

0 √

√ 1

Tan 0 √

(^1) √ U D (^) √ - 1 √

0 √

(^1) √ U D (^) √ - 1 √

0

UD- Undefined TRIGONOMETRIC IDENTITIES

𝑥 𝑥 𝑥^ 𝑥

RECIPROCAL IDENTITIES

PYTHAGOREAN IDENTITIES

What’s New

Many animal populations, such as that of rabbits, fluctuate over ten- year cycles. Suppose that the number of rabbits at time t (in years) is given by

Question:

For what value values of t does the rabbit population exceed 4500 when

The answer for this problem is between 0 years to years ( )

and from up to years ( ).

In the proceeding lesson, you will be learning the basics of solving trigonometric equations and solve situational problems that involve trigonometric equations.

What is It

A. SOLVING TRIGONOMETRIC EQUATION

When you are trying “to solve an equation”, this means that you will find all the solutions that will make the equation true. Here, unless stated as angles measured in degrees, we mean solutions of the equations that are real numbers (or equivalently, angles measured in radian.

Before going to the first example, take note of the following tips in solving trigonometric equations:

1. If the equation contains only one trigonometric term, isolate that term, and solve for the variable.
2. If the equation is quadratic in form, we may use factoring, finding the square roots, or the quadratic formula.

3. Rewrite the equation to have 0 on one side, and then factor (if appropriate) the expression on the other side.
4. If the equation contains mote that one trigonometric function, try to express everything in terms of one trigonometric function. Here, identities are useful.
5. If half or multiple angles are present, express them in terms of a trigonometric expression of a single angle, except when all angles involved have the same multiplicity wherein, in this case, retain the angle. Half-angle and double- angle identities are useful in simplification.

A.1 Equations with one term

Example 1. Solve the equation. Solution: Given Add 1 to both sides Divide both sides with 2 Equivalent equation Solve for x and Using the table of trigonometric values or solving with scientific calculator in radian Note:  For , the solution of are (this is in Q and (in Q2).  Any angle that is co terminal with or will also be a solution of the equation.  Because the period of sine and cosine function is , the complete solution of the equation are and , for all integer k.

Example 4. Solve for. Solution Given Add on both sides Sine Double-angle Identity Factor

, , x , Solutions: 0, , , ,

Example 5. Solve for. Solution Given Pythagorean Identity Distribution; Add on both sides Combine like terms Divide/Multiply both sides by - Factoring Quadratic Solve each Factor

x Values from Trigo table or by , using a scientific calculator

Solutions: , ,

Example 6. Solve

Solution:

( )

• ( )+

x x x no solution x , x

Solutions : x , x

B. SOLVING SITUATIONAL PROBLEM INVOLING TRIGONOMETRIC

EQUATIONS

Example 7. A weight is suspended from a spring and vibrating vertically according to the equation

( ( ))

where centimetres is the directed distance of the weight form the central position at seconds, and the positive distance means above its central position. (1) At what time is the displacement of the weight 5 cm below its central position for the first time? (2) For what values of does the weight reach its farthest point below its central position?

Here, the least positive value of happens when and this is

(2) The minimum value happens when and only when the minimum

value of ( ) is reached. The minimum value of ( ) is - 1, which implies that the farthest point the weight can reach below its central position is 20 cm. Thus, we need to solve for all values of such that ( ) ( ) ( )

( ) ( )

Therefore, the weight reaches its farthest point (which is 20 cm)

below its central position at for every integer

What’s More

A. In this activity you will be required to supply the missing step or data in solving the trigonometric equation below.

1. Solve:. Solution:

, Solutions: , +

2. If + , Solve the equation: (^) √ Solution: √

√

√

, Solutions:

b.

a.

c.

f.

e. d.

g. i.

h. j.

k.

l.

What I Have Learned

Based on the concepts that you learn from this module, complete all the following sentences.

1. The equation that contains trigonometric expressions is called __________.
2. To get the complete solution of trigonometric equation involving sin and cosine, you need to add _________________ since sin and cosine functions have two period intervals.
3. To get the complete solution of trigonometric equation involving tangent function you need to add _________________ since tangent functions have one period interval.
4. When the equation is in quadratic form, we can use __________________, _______________________, __________________ to solve for the variable.

What I Can Do

This section involves real-life application of the concept of solving trigonometric equations.

1. Cite a problem that involves a trigonometric equation. Show its complete solution and share/ post it in a social media sites such as facebook, instagram, etc. Solution:

Assessment

Multiple Choice. Choose the letter of the best answer. Write the chosen letter on a separate sheet of paper.

16. What value of will satisfy the trigonometric equation √ a. c. b. d.
17. How many possible solutions does. a. c. b. d.
18. All are possible solutions of a. c. b. d.
19. What value of x is true to all 3 trigonometric equations: √ , and a.. c. b. d.
20. has an equivalent equation of ___________ a. ( c. ( b. ( d. (
21. has a complete solution of ________________.

a. , c. ,

b. , d. ,

22. If is the current (in amperes) in an alternating current circuit at time (in seconds), find the smallest exact value of for which if and. a. c. b. d.
23. One of the following choices is a solution of the trigonometric equation =0. Which one is it? a. c. b. d.