2.5 Trigonometric Functions

Why This Section Matters

Trigonometric functions on Praxis 5165 emphasize the unit circle, reference angles, and solving basic equations on a stated interval such as 0 ≤ x < 2π. You need exact values at special angles and quadrant sign rules. The on-screen calculator helps on test day, but many items expect symbolic reasoning without decimal approximations.

The Unit Circle

A point at angle θ (radians, measured from the positive x-axis) on the unit circle has coordinates (cos θ, sin θ).

Memorize exact values:

Reference Angles and ASTC Signs

The reference angle is the acute angle between the terminal side and the x-axis.

ASTC quadrant mnemonic — which ratios are positive:

• Q I: All positive
• Q II: Sin positive
• Q III: Tan positive
• Q IV: Cos positive

Worked Example: cos(5π/3)

5π/3 lies in Quadrant IV (between 3π/2 and 2π).

Reference angle: 2π - 5π/3 = π/3

Cosine is positive in Q IV, so cos(5π/3) = cos(π/3) = 1/2.

Worked Example: Solve sin x = √3/2 on 0 ≤ x < 2π

Reference angle where sin = √3/2 is π/3.

Sine is positive in Q I and Q II:

x = π/3 or x = 2π/3

Always list all solutions in the given interval — Praxis distractors often include only one angle.

Graphs: Amplitude and Period

y = sin x and y = cos x have period 2π and amplitude 1.

For y = a sin(bx):

• Amplitude = |a|
• Period = 2π/|b|

Example: y = 3 sin(2x) has amplitude 3 and period π.

Phase shifts appear in transformation items: y = sin(x - π/4) shifts the sine wave right π/4.

Radians and Degrees

Conversions you must execute quickly:

• degrees = radians × (180/π)
• radians = degrees × (π/180)

5π/3 radians = 300°. 210° = 7π/6 radians.

Praxis 5165 mixes both unit systems; convert before applying quadrant reasoning if needed.

Inverse Trig (Conceptual)

sin⁻¹(x) returns an angle in a restricted range whose sine is x. Praxis items rarely require heavy inverse-trig algebra, but know inverses undo trig functions with domain/range restrictions.

Right-Triangle Connection

For acute angle θ in a right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Unit-circle values extend these ratios to all quadrants via reference angles.

Common Praxis Traps

• Wrong quadrant sign (negative cosine in Q IV when magnitude is 1/2)
• Missing the second solution on 0 ≤ x < 2π
• Swapping π/6 and π/3 values
• Reporting the reference angle instead of the actual solution angle

Section Takeaways

Use reference angles plus ASTC signs for exact evaluations; list every solution in the interval. Drill unit-circle and trig-equation items on /practice/praxis-math.

The Pythagorean Identity

sin²θ + cos²θ = 1 follows from the unit circle equation x² + y² = 1 with x = cos θ, y = sin θ.

Use it to find sin θ when cos θ is known (and quadrant is specified):

If cos θ = -3/5 in Q II, then sin²θ = 1 - 9/25 = 16/25, so sin θ = 4/5 (positive in Q II).

Tangent and Cotangent

tan θ = sin θ / cos θ (where cos θ ≠ 0).

For θ = π/3: tan(π/3) = (√3/2)/(1/2) = √3.

cot θ = 1/tan θ. Know where tangent is undefined — odd multiples of π/2.

Graphing Sine and Cosine Shifts

y = cos(x - π/6) shifts the cosine graph right π/6. The first maximum after the origin occurs at x = π/6 instead of x = 0.

Compare to y = cos x + 2, which shifts up 2 without changing period.

Special Angle Practice Grid

Memorize this grid; Praxis items often test two entries and expect you to deduce the rest via identities.

Teaching Angle Measure

When students mix degrees and radians, assign both labels: π/6 = 30°. Building a unit-circle diagram with degree and radian ticks prevents Quadrant IV sign errors on cosine items like cos(5π/3).

Cofunction Identities

sin(π/2 - θ) = cos θ and cos(π/2 - θ) = sin θ.

Example: sin(π/6) = cos(π/3) = 1/2. These identities help when angles are given as complements.

Solving cos x = -1/2

On 0 ≤ x < 2π, reference angle π/3 where cos = 1/2. Cosine is negative in Q II and Q III:

x = 2π/3 or x = 4π/3.

Listing both distinguishes complete solution sets from partial answers.

Unit Circle Symmetry

cos(-θ) = cos θ (even function) sin(-θ) = -sin θ (odd function)

Use symmetry to find values in negative-angle or clockwise contexts without recomputing from scratch.

Evaluating sin(5π/6) Step by Step

5π/6 is in Quadrant II. Reference angle: π - 5π/6 = π/6.

Sine is positive in Q II, so sin(5π/6) = sin(π/6) = 1/2.

Parallel reasoning applies to any obtuse angle: locate quadrant, find reference angle, apply ASTC sign.

Periodicity

sin(x + 2π) = sin x and cos(x + 2π) = cos x. When solving on 0 ≤ x < 2π, stop after finding all solutions in one full revolution; when the interval is wider, add 2πk for integer k.

Link to Praxis 5165 Practice

Trigonometry items often appear alongside transformation and function-notation questions in mixed timed sets. After reading this section, filter /practice/praxis-math for exponential-logarithmic-trig and function-notation topics to rehearse special-angle fluency under exam pacing.