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