Trigonometric Equations Practice Questions

Trigonometric Equations are used to calculate the angles that meet a specific trigonometric relationship. These equations are essential in fields like physics, engineering, and navigation, where accurate angle measurements are critical.

General Solutions for Trigonometric Equations

Trigonometrical Equations	Solutions
sin θ = 0	θ = nπ
cos θ = 0	θ = (nπ + π/2)
tan θ = 0	θ = nπ
sin θ = 1	θ = (2nπ + π/2) = (4n+1) π/2
cos θ = 1	θ = 2nπ
tan θ = 1	θ = nπ + π/4
sin θ = sin α	θ = nπ + (-1)ⁿ α, where α ∈ [-π/2, π/2]
cos θ = cos α	θ = 2nπ ± α, where α ∈ (0, π]
tan θ = tan α	θ = nπ + α, where α ∈ (-π/2, π/2]
sin 2θ = sin 2α	θ = nπ ± α
cos 2θ = cos 2α	θ = nπ ± α
tan 2θ = tan 2α	θ = nπ ± α

Trigonometric Equations: Practice Questions with Solution

Question 1: Solve for θ if sin (θ) = 1/2

Solution:

To solve for θ given that
sin(θ) = 1/2
We need to find the angles where the sine function equals 1/2.
Sine function equals 1/2 at specific angles within the standard range of o to 360° ( 0 to 2π radians):
sin(θ) = 1/2
θ = 30°, and 150° (in degrees)
θ = π/6, 5π/6 ( in radians)

Question 2: Solve for θ if cos (θ) = -1/2.

Solution:

To solve for θ given that
cos(θ) = -1/2,
We need to find the angles where the sine function equals -1/2.
θ = arccos (-1/2) = 120° or 240°
⇒ θ = 120° + 360°(n) or 240°+ 360°(n), where n = 1, 2, . . .

Question 3: Solve for θ if tan(θ) = 1

Solution:

To solve for θ given that
tan(θ) = 1,
we need to find the angles where the sine function equals 1.
θ = arctan (1) = 45° or 225°
⇒ θ = 45° + 360°(n), where n = 1, 2, . . .

Question 4: Solve for θ: 2cos²θ - 1 = 0.

Solution:

2cos²θ - 1 = 0
⇒ cos²θ = 1/2
⇒ cos θ = ± 1/√2
⇒ θ = 45^o, 135^o, 225^o, 315^o

Question 5: Solve 2sin(θ) + 1 = 0 for 0 ≤ θ ≤ 2π.

Solution:

2sin(θ) + 1 = 0
⇒ sin(θ) = -1/2
⇒ θ = 7π/6, 11π/6

Question 6: Solve sin(θ) cos(θ) = 1/4 for 0 ≤ θ ≤ 2π.

Solution:

First, use the double-angle identity for sine:
sin(θ) cos(θ) = (1/2) sin(2θ)
Thus, the equation becomes:(1/2) sin(2θ) = 1/4
Multiply both sides by 2 to solve for sin(2θ):
sin(2θ) = 1/2
Sine function has a value of 1/2 at the angles:
2θ = π/6, 5π/6, 13π/6, 17π/6
⇒ θ = π/12, 5π/12, 13π/12, 17π/12
Thus, the solution to the equation sin(θ) cos (θ) = 1/4 for 0 ≤ θ ≤ 2 are:
θ = π/12, 5π/12, 13π/12, 17π/12

Question 7: Solve tan(2θ) = √3 for 0 ≤ θ ≤ 2π

Solution:

First, solve for 2θ in the equation tan(2θ) = √3 .
Tangent function has a value of √3 at the angles: .
2θ = π/3, 4π/3
⇒ θ = π/6, 2π/3, 7π/6, 5π/3
Thus, the solutions to the equation tan(2θ) = √3 for 0 ≤ θ ≤ 2π are:
θ = π/6, 2π,/3, 7π/6, 5π/3

Question 8: Solve cos(3θ) = 1 for 0 ≤ θ < 2π.

Solution:

First, solve for 3θ in the equation cos(3θ) = 1.
Cosine function has a value of 1 at the angles:
3θ = 0, 2π, 4π, 6π
⇒ θ = 0, 2π/3 , 4π/3
Thus, the solution to the equation cos(3θ) = 1 for 0 ≤ θ ≤ 2π are:θ = 0, 2π/3, 4π/3

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Trigonometric Equations: Worksheet

The worksheet on trigonometric equation is added in form of image below:

Worksheet_-Trigonometric-Equations-1---Google-Docs_page-0001

Answer Key

θ = 4π/3, 5π/3
θ = π/4, 3π/4, 5π/4, π
θ = π/3, 4π/3
θ = 0, π/3, 2π/3, π, 4π/3
θ = π/4, 7π/4
θ = π/3, 2π/3, 4π/3, π
θ = π/6, 5π/6
θ = 0, π/4, π/2, 3π/4, π, 5π/4, 3π/2, 7π/4
θ = π/6, 7π/6, 11π/6
θ = 3π/8, 7π/8, 11π/8, 15π/8

Trigonometric Equations Practice Questions

General Solutions for Trigonometric Equations

​Trigonometric Equations: Practice Questions with Solution

Trigonometric Equations: Worksheet

Answer Key

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Trigonometric Equations: Practice Questions with Solution