Find all solutions to the equation.3sinθ−6=−93\sin\theta-6=-93sinθ−6=−9

θ=3π2+2πn\theta=\frac{3\pi}{2}+2\pi nθ=23π+2πn

Find all solutions to the equation. 3 sin θ − 6 = − 9 3\sin\theta-6=-9 3 sin θ − 6 = − 9

This problem, we're asked to find all solutions to the equation 3 sine theta minus 6 is equal to negative 9. Now we're going to start with step number 1 of how to solve linear trig equations. So we want to go ahead and isolate our trig function using whatever operations I need to. So here, my trig function is the sine of theta. That's what I'm aiming to isolate. So the first step in doing that is going to be to add 6 to both sides. So that will cancel out, leaving me with 3 sine of theta is equal to negative 9 plus 6, which is negative 3. Now from here to isolate that sine theta, I can just divide both sides by 3. Then that 3 will cancel, leaving me with sine of theta is equal to negative 3 divided by positive 3, which is going to give me negative 1. So step number 1 is done here, and we can go ahead and move on to step number 2, which is to find all of our solutions on the unit circle. So we want to find every angle for which the sine is equal to negative one. So coming down here to my unit circle looking for where my sine is equal to negative one, I know that down here at 3 pi over 2, that's where my sine is going to be. That's negative one. So here I get that theta is equal to 3 pi over 2. Now there are no other angles on my unit circle for which that is true, so that represents all of my solutions that are on that unit circle. So I have completed step number 2. Now for step number 3, we are told that if our domain is not restricted, we need to add 2pyn to each solution. So always remember to double check what you're asked to find in the problem. In this case, our problem asked us to find all of the solutions, so that means that our domain is not restricted and we do want to add 2 pi in here. So step number 3 is done, and then step number 4 is to isolate theta. Now theta is already by itself and completely isolated here, so step number 4 is done as well. And we have our final solution, which is 3pitwo+2pitin. That represents all solutions to the given equation. Thanks for watching, and let me know if you have any questions.

Find all solutions to the equation.3sin⁡θ−6=−93\sin\theta-6=-93sinθ−6=−9

θ=3π2+2πn\theta=\frac{3\pi}{2}+2\pi nθ=23π+2πn

θ=π2+2πn\theta=\frac{\pi}{2}+2\pi nθ=2π+2πn

θ=π+2πn\theta=\pi+2\pi nθ=π+2πn

Skip topic navigation

0. Fundamental Concepts of Algebra3h 32m

Algebraic Expressions
37m

Exponents
32m

Polynomials Intro
19m

Multiplying Polynomials
36m

Factoring Polynomials
2m

Radical Expressions
15m

Simplifying Radical Expressions
31m

Rationalize Denominator
15m

Rational Exponents
4m

Pythagorean Theorem & Basics of Triangles
17m

1. Equations and Inequalities3h 27m

Linear Equations
31m

Rational Equations
21m

The Imaginary Unit
6m

Powers of i
11m

Complex Numbers
42m

Intro to Quadratic Equations
18m

The Square Root Property
11m

Completing the Square
12m

The Quadratic Formula
19m

Choosing a Method to Solve Quadratics
9m

Linear Inequalities
22m

2. Graphs1h 43m

Graphs and Coordinates
7m

Two-Variable Equations
23m

Lines
1h 12m

3. Functions & Graphs2h 17m

Intro to Functions & Their Graphs
35m

Common Functions
5m

Transformations
45m

Function Operations
21m

Function Composition
29m

4. Polynomial Functions1h 54m

Quadratic Functions
49m

Understanding Polynomial Functions
34m

Graphing Polynomial Functions
30m

5. Rational Functions1h 23m

Introduction to Rational Functions
9m

Asymptotes
35m

Graphing Rational Functions
38m

6. Exponential and Logarithmic Functions2h 28m

Introduction to Exponential Functions
9m

Graphing Exponential Functions
25m

The Number e
8m

Introduction to Logarithms
22m

Graphing Logarithmic Functions
18m

Properties of Logarithms
27m

Solving Exponential and Logarithmic Equations
35m

7. Measuring Angles40m

Angles in Standard Position
12m

Coterminal Angles
8m

Complementary and Supplementary Angles
10m

Radians
8m

8. Trigonometric Functions on Right Triangles2h 5m

Trigonometric Functions on Right Triangles
45m

Special Right Triangles
30m

Cofunctions of Complementary Angles
26m

Solving Right Triangles
23m

9. Unit Circle1h 19m

Defining the Unit Circle
14m

Trigonometric Functions on the Unit Circle
9m

Common Values of Sine, Cosine, & Tangent
11m

Reference Angles
38m

Reciprocal Trigonometric Functions on the Unit Circle
6m

10. Graphing Trigonometric Functions1h 19m

Graphs of the Sine and Cosine Functions
32m

Phase Shifts
14m

Graphs of Secant and Cosecant Functions
10m

Graphs of Tangent and Cotangent Functions
21m

11. Inverse Trigonometric Functions and Basic Trig Equations1h 41m

Inverse Sine, Cosine, & Tangent
28m

Evaluate Composite Trig Functions
44m

Linear Trigonometric Equations
28m

12. Trigonometric Identities 2h 34m

Introduction to Trigonometric Identities
1h 4m

Sum and Difference Identities
1h 3m

Double Angle Identities
16m

Solving Trigonometric Equations Using Identities
9m

13. Non-Right Triangles1h 38m

Law of Sines
49m

Law of Cosines
30m

Area of SAS & ASA Triangles
19m

14. Vectors2h 25m

Geometric Vectors
28m

Vectors in Component Form
32m

Direction of a Vector
23m

Unit Vectors and i & j Notation
27m

Dot Product
22m

Cross Product
11m

15. Polar Equations2h 5m

Polar Coordinate System
29m

Convert Points Between Polar and Rectangular Coordinates
26m

Convert Equations Between Polar and Rectangular Forms
29m

Graphing Other Common Polar Equations
39m

16. Parametric Equations1h 6m

Graphing Parametric Equations
12m

Eliminate the Parameter
32m

Writing Parametric Equations
21m

17. Graphing Complex Numbers1h 7m

Graphing Complex Numbers
6m

Polar Form of Complex Numbers
22m

Products and Quotients of Complex Numbers
14m

Powers of Complex Numbers (DeMoivre's Theorem)
23m

18. Systems of Equations and Matrices3h 6m

Two Variable Systems of Linear Equations
8m

Introduction to Matrices
1h 11m

Determinants and Cramer's Rule
1h 3m

Graphing Systems of Inequalities
42m

19. Conic Sections2h 36m

Introduction to Conic Sections
6m

Circles
26m

Ellipses: Standard Form
33m

Parabolas
36m

Hyperbolas at the Origin
40m

Hyperbolas NOT at the Origin
12m

20. Sequences, Series & Induction1h 15m

Sequences
33m

Arithmetic Sequences
25m

Geometric Sequences
15m

21. Combinatorics and Probability1h 45m

Factorials
11m

Combinatorics
46m

Probability
47m

22. Limits & Continuity1h 49m

Introduction to Limits
41m

Finding Limits Algebraically
40m

Continuity
27m

23. Intro to Derivatives & Area Under the Curve2h 9m

Tangent Lines & Derivatives
44m

Limits at Infinity
18m

Limits of Sequences
9m

Area Under a Curve
56m

11. Inverse Trigonometric Functions and Basic Trig Equations

Linear Trigonometric Equations

Precalculus

11. Inverse Trigonometric Functions and Basic Trig Equations

Linear Trigonometric Equations

Previous problemNext problem

Struggling with Precalculus?

Join thousands of students who trust us to help them ace their exams!

Multiple Choice

Find all solutions to the equation.
$3$\sin\]\theta$-6=-9$

$$\theta$=$\frac{\pi}{2}$+2$\pi$ n$

$$\theta$=2$\pi$ n$

$$\theta$=$\pi$+2$\pi$ n$

$$\theta$=$\frac{3\pi}{2}$+2$\pi$ n$

0 댓글

Previous problemNext problem

검증된 단계별 안내

Start by isolating the sine function in the equation 3sin(θ) - 6 = -9. Add 6 to both sides to get 3sin(θ) = -3.

Divide both sides of the equation by 3 to solve for sin(θ), resulting in sin(θ) = -1.

Recall that the sine function equals -1 at specific angles. The general solution for sin(θ) = -1 is θ = 3π/2 + 2πn, where n is an integer.

Verify the solution by substituting θ = 3π/2 + 2πn back into the original equation to ensure it satisfies 3sin(θ) - 6 = -9.

Conclude that the solutions to the equation are given by θ = 3π/2 + 2πn, where n is any integer, representing all angles coterminal with 3π/2.

3:29m

Watch next

Master Example 3 with a bite sized video explanation from Patrick Ford

Linear Trigonometric Equations practice set

Problem sets built by lead tutors

Expert video explanations

Struggling with Precalculus?

Find all solutions to the equation.3sinθ−6=−93\(\sin\]\theta\)-6=-93sinθ−6=−9

Watch next

Linear Trigonometric Equations practice set

Find all solutions to the equation.
$3\(\sin\]\theta\)-6=-9$