Study guide
5.3 Trigonometry Basics
ALEKS Math Placement Study Guide Exam
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1 Chapter 1: ALEKS PPL Overview & Strategy
1.1 What ALEKS PPL Is 1.2 How Placement Works 1.3 The Prep & Learning Module & Retake Strategy
2 Chapter 2: Arithmetic & Number Sense
2.1 Real Numbers & Operations 2.2 Fractions, Decimals & Percents 2.3 Ratios, Proportions & Unit Conversion
3 Chapter 3: Algebra Essentials
3.1 Linear Equations & Inequalities 3.2 Exponents & Polynomials 3.3 Factoring & Rational Expressions 3.4 Quadratic Equations
4 Chapter 4: Functions, Graphs & Geometry
4.1 Lines, Slope & Graphing 4.2 Functions & Function Notation 4.3 Geometry & Measurement 4.4 Coordinate Geometry
5 Chapter 5: Advanced Readiness — Precalculus & Trig
5.1 Exponential & Logarithmic Expressions 5.2 Radicals, Rational Exponents & Complex Numbers 5.3 Trigonometry Basics 5.4 Sequences, Functions Review & Exam Readiness
5.3 Trigonometry Basics
Key Takeaways Explain This Topic AI
SOH-CAH-TOA: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent for a right triangle.
Convert degrees to radians by multiplying by π/180, and radians to degrees by multiplying by 180/π; 180° = π radians.
Special angles 30°, 45°, 60° have exact values worth memorizing: sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2.
The Pythagorean identity sin²θ + cos²θ = 1 holds for every angle and lets you find one ratio from another.
On the unit circle a point is (cos θ, sin θ); at 90° that is (0, 1), so cos 90° = 0 and sin 90° = 1.
Last updated: June 2026
Right-Triangle Trig: SOH-CAH-TOA
For a right triangle with an acute angle θ, label the sides relative to θ: the hypotenuse is opposite the right angle, the opposite side faces θ, and the adjacent side touches θ. The three primary ratios are remembered by SOH-CAH-TOA :
- sin θ = Opposite / Hypotenuse
- cos θ = Adjacent / Hypotenuse
- tan θ = Opposite / Adjacent
Worked example. A right triangle has legs 3 and 4 and hypotenuse 5 (a 3-4-5 triangle). For the angle θ opposite the side of length 3: sin θ = 3/5 = 0.6, cos θ = 4/5 = 0.8, and tan θ = 3/4 = 0.75.
Worked example — find a side. A ramp makes a 30° angle with the ground and is 10 ft long (hypotenuse). Its height is the opposite side: sin 30° = height/10, and sin 30° = 1/2, so height = 10·(1/2) = 5 ft .
Worked example — find an angle. A right triangle has opposite side 7 and adjacent side 7. Then tan θ = 7/7 = 1, and the angle whose tangent is 1 is θ = 45° (the inverse, written θ = tan⁻¹(1) = 45°). The three reciprocal ratios also exist: cosecant csc θ = 1/sin θ, secant sec θ = 1/cos θ, and cotangent cot θ = 1/tan θ. ALEKS occasionally asks for these, but they are just reciprocals of the main three.
Common mistake: "Opposite" and "adjacent" are defined relative to the chosen angle , not fixed sides. Re-label every time you switch reference angles.
Degrees and Radians
Angles are measured in degrees (a full circle is 360°) or radians (a full circle is 2π). The bridge is 180° = π radians , which gives the conversions:
Degrees → radians: multiply by π/180 .
Radians → degrees: multiply by 180/π .
Worked example. Convert 60° to radians: 60·(π/180) = π/3 radians.
Worked example. Convert 3π/4 radians to degrees: (3π/4)·(180/π) = 3·45 = 135°.
The Unit Circle and Special Angles
The unit circle is a circle of radius 1 centered at the origin. For an angle θ measured from the positive x-axis, the point on the circle is (cos θ, sin θ) . So at θ = 0° the point is (1, 0); at θ = 90° it is (0, 1), giving cos 90° = 0 and sin 90° = 1; at 180° it is (−1, 0).
The three special acute angles — 30°, 45°, 60° — have exact values that ALEKS expects you to know cold. Here is the reference table:
- θ (deg) θ (rad) sin θ cos θ tan θ
- 0° 0 0 1 0
- 30° π/6 1/2 √3/2 √3/3
- 45° π/4 √2/2 √2/2 1
- 60° π/3 √3/2 1/2 √3
- 90° π/2 1 0 undefined
Notice the symmetry: sin and cos values are mirror images (sin 30° = cos 60°), and tan = sin/cos. tan 90° is undefined because cos 90° = 0 (division by zero).
Evaluating at Special Angles
Worked example. Evaluate tan 45°. From the table sin 45° = √2/2 and cos 45° = √2/2, so tan 45° = (√2/2)/(√2/2) = 1 .
Worked example. Find cos 60° + sin 30°. Both equal 1/2, so the sum is 1/2 + 1/2 = 1 .
Worked example — a triangle problem. A guy-wire runs from the top of a 60-ft pole to the ground, making a 60° angle with the ground. How long is the wire? The pole height (60 ft) is opposite the 60° angle and the wire is the hypotenuse: sin 60° = 60/wire, so wire = 60/sin 60° = 60/(√3/2) = 120/√3 = 40√3 ≈ 69.3 ft.
The Pythagorean Identity
The most important identity comes straight from the unit circle, where x² + y² = 1 with x = cos θ and y = sin θ:
sin²θ + cos²θ = 1
This holds for every angle and lets you find one ratio when you know the other.
Worked example. If sin θ = 3/5 and θ is acute, find cos θ. Substitute: (3/5)² + cos²θ = 1 ⟹ 9/25 + cos²θ = 1 ⟹ cos²θ = 16/25 ⟹ cos θ = 4/5 (positive because θ is acute). This recovers the 3-4-5 triangle.
Worked example. Verify the identity at 30°: sin²30° + cos²30° = (1/2)² + (√3/2)² = 1/4 + 3/4 = 1 . ✓
Worked example — using the identity to evaluate. Suppose cos θ = −4/5 with θ in the second quadrant (where sine is positive). Then sin²θ = 1 − 16/25 = 9/25, so sin θ = +3/5 (positive because of the quadrant). The identity gives the magnitude; the quadrant fixes the sign.
The unit circle extends these ratios beyond acute angles. Angles measured counterclockwise from the positive x-axis sweep through four quadrants , and the signs of sin and cos follow the point's coordinates: in Quadrant I both are positive; in Quadrant II cos is negative; in Quadrant III both are negative; in Quadrant IV sin is negative. A common memory aid is "All Students Take Calculus" for which ratio is positive in each quadrant (All, Sine, Tangent, Cosine).
Common mistake: sin²θ means (sin θ)², not sin(θ²). And the identity is sin² + cos² = 1, not sin + cos = 1 — squaring is essential. A quick sanity check: at 45°, √2/2 + √2/2 = √2 ≈ 1.41, which is clearly not 1, but the squares sum to 1.