5.3 Trigonometry Basics

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:

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.