Solving Trigonometric Equations (Linear)

Find coterminal and reference angles for any angle. Grade 12 Math 30-1.

Lesson 4.4 of Trigonometric Ratios & Functions in Math 30-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Solving First Degree Trigonometric Equations. Master the 4-step process to solve first degree trig equations using the unit circle. Learn to find reference angles, determine quadrants using CAST, and handle special cases with secant, cosecant, and cotangent.
• General Solution. Learn how to write general solutions for trigonometric equations when the domain is infinite. Understand the notation θ + 360°n or θ + 2πn, and discover special cases for symmetrically apart solutions.

Inside the lesson — a free preview

Concept

A first degree trigonometric equation contains a trig ratio (sin, cos, tan, etc.) with the variable inside the angle.

Example:. θ = -(√(3))/(2)

To solve a trig equation, we use our knowledge of exact trig values from the unit circle.

Steps for Solving

Step 1: Isolate the trig ratio. Get the trigonometric function by itself on one side.

Example:

θ = -(√(3))/(2)

Step 2: Use the unit circle to find angles. Look at the unit circle and find which angles have that exact coordinate value.

Remember:

• For sine: look at the y-coordinate

• For cosine: look at the x-coordinate

• For tangent: look at y/x

For θ = -(√(3))/(2), look for points where the y-coordinate is -(√(3))/(2).

Since the value is negative, the y-coordinate is below the x-axis. This happens in:

• Quadrant III (x negative, y negative)

• Quadrant IV (x positive, y negative)

From the unit circle, this occurs at:

θ = (4π)/(3) and (5π)/(3)

Step 3: Find all angles in the given domain. If the domain is larger than 0 to 2π, add or subtract full rotations (2π or 360°) to find all solutions.

Example with specific domain:

For 0 ≤ θ ≤ 2π:

θ = (4π)/(3), (5π)/(3)

For 0 ≤ θ ≤ 4π:

θ = (4π)/(3), (5π)/(3), (10π)/(3), (11π)/(3)

When a Different Trig Function Appears

Sometimes equations contain secant, cosecant, or cotangent.

Rewrite them using sine, cosine, or tangent.

Example:. θ = √(2)

Since θ = (1)/( θ), we rewrite as:

θ = (1)/(√(2))

Rationalize the denominator:. Multiply by (√(2))/(√(2)):

θ = (1)/(√(2)) · (√(2))/(√(2)) = (√(2))/(2)

Then solve normally.

From the unit circle, the x-coordinate gives the cosine value.

θ = (√(2))/(2)

occurs at reference angle _r = (π)/(4)

Cosine is positive in Quadrants I and IV.

If the domain is 0 ≤ θ ≤ 2π, the solutions are:

θ = (π)/(4), (7π)/(4)

Worked example: Example 1

Solve θ = -√(3), 0° ≤ θ ≤ 360°

Solution:. Step 1: Set up the equation

On the unit circle, θ = (y)/(x)

We need: (y)/(x) = -√(3)

Step 2: Find coordinates where y/x = -√3

Look at the unit circle. We need to find points (x, y) where:

(y)/(x) = -√(3)

This means y = -√(3) · x

For tangent to be negative, x and y must have opposite signs.

Step 3: Check the unit circle coordinates

Looking at the unit circle, we find:

• Coordinates: (-(1)/(2), (√(3))/(2))

Check: (y)/(x) = ((√(3))/(2))/(-(1)/(2)) = -√(3) ✓

• Coordinates: ((1)/(2), -(√(3))/(2))

Check: (y)/(x) = (-(√(3))/(2))/((1)/(2)) = -√(3) ✓

Step 4: Identify what angles these coordinates correspond to

From the unit circle:

• The point (-(1)/(2), (√(3))/(2)) is at 120°

• The point ((1)/(2), -(√(3))/(2)) is at 300°

Final Answer:

θ = 120°, 300°

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