Pre-Calculus - Week 9

Right Triangle Trigonometry

Trigonometry begins with triangles. The six trig functions — sine, cosine, tangent, and their reciprocals — are defined as ratios of sides in a right triangle. These ratios are the same for any similar triangle with the same angle, making them a universal tool for measuring inaccessible distances, engineering structures, analyzing waves, and solving physics problems. Calculus depends on them at every turn.

Learning Objectives

- Define the six trigonometric ratios using SOH-CAH-TOA. - Evaluate trig functions for 30°, 45°, and 60° from memory. - Use a calculator to evaluate trig functions for any angle. - Apply the Pythagorean identities. - Solve right triangles given two pieces of information. - Apply trig to angle of elevation and depression problems. - Define reference angles for angles in any quadrant.

1. The Six Trigonometric Ratios

For a right triangle with angle θ (not the right angle), let: - Opposite = side opposite θ - Adjacent = side next to θ (not the hypotenuse) - Hypotenuse = side opposite the right angle (always longest) SOH-CAH-TOA: sin(θ) = Opposite / Hypotenuse cos(θ) = Adjacent / Hypotenuse tan(θ) = Opposite / Adjacent Reciprocals: csc(θ) = 1/sin(θ) = Hyp/Opp sec(θ) = 1/cos(θ) = Hyp/Adj cot(θ) = 1/tan(θ) = Adj/Opp

2. Example: Finding All Six Ratios

A right triangle has legs 3 and 4 and hypotenuse 5 (a 3-4-5 triangle). Angle θ is opposite the leg of length 4. sin(θ) = 4/5 cos(θ) = 3/5 tan(θ) = 4/3 csc(θ) = 5/4 sec(θ) = 5/3 cot(θ) = 3/4 Check: sin^2(θ) + cos^2(θ) = 16/25 + 9/25 = 25/25 = 1. ✓

3. Special Angles: 30°, 45°, 60°

These angles appear constantly. Know them cold: 45°-45°-90° triangle: legs = 1, 1; hypotenuse = sqrt(2). sin(45°) = 1/sqrt(2) = sqrt(2)/2 cos(45°) = sqrt(2)/2 tan(45°) = 1 30°-60°-90° triangle: short leg = 1, long leg = sqrt(3), hypotenuse = 2. sin(30°) = 1/2, cos(30°) = sqrt(3)/2, tan(30°) = 1/sqrt(3) = sqrt(3)/3 sin(60°) = sqrt(3)/2, cos(60°) = 1/2, tan(60°) = sqrt(3) Memory tip: sin increases from 30° to 60°: sin(30°) = sqrt(1)/2, sin(45°) = sqrt(2)/2, sin(60°) = sqrt(3)/2.

4. The Pythagorean Identities

These come directly from sin^2 + cos^2 = 1 (Pythagorean theorem on a unit circle). Identity 1: sin^2(θ) + cos^2(θ) = 1 Identity 2: 1 + tan^2(θ) = sec^2(θ) Identity 3: 1 + cot^2(θ) = csc^2(θ) Identities 2 and 3 are derived by dividing Identity 1 by cos^2(θ) or sin^2(θ). Example: If sin(θ) = 3/5, find cos(θ) (assuming θ in first quadrant). cos^2(θ) = 1 - sin^2(θ) = 1 - 9/25 = 16/25. cos(θ) = 4/5.

5. Co-function Identities

For complementary angles (sum = 90°): sin(θ) = cos(90° - θ) cos(θ) = sin(90° - θ) tan(θ) = cot(90° - θ) Example: sin(30°) = cos(60°) = 1/2. ✓ Example: tan(20°) = cot(70°). This is why the prefix 'co' exists in cosine, cotangent, cosecant — each is the function of the complementary angle.

6. Solving a Right Triangle

To solve a right triangle means finding all unknown sides and angles. Given: angle A = 35°, hypotenuse c = 20. Find sides a and b. sin(35°) = a/20 → a = 20*sin(35°) ≈ 20 * 0.5736 ≈ 11.47. cos(35°) = b/20 → b = 20*cos(35°) ≈ 20 * 0.8192 ≈ 16.38. Angle B = 90° - 35° = 55°. Given: legs a = 5 and b = 12. c = sqrt(25 + 144) = sqrt(169) = 13. tan(A) = 5/12. A = arctan(5/12) ≈ 22.6°. B = 90° - 22.6° = 67.4°.

7. Angle of Elevation and Depression

Angle of elevation: measured upward from horizontal to a line of sight above. Angle of depression: measured downward from horizontal to a line of sight below. These angles are equal (alternate interior angles with parallel horizontal lines). Example 1: From 50 m away, you look up at the top of a tower at 72° elevation. Find the tower height. tan(72°) = h/50. h = 50*tan(72°) ≈ 50 * 3.0777 ≈ 153.9 m. Example 2: A pilot at 3000 ft sees a runway at a 15° angle of depression. Find horizontal distance. tan(15°) = 3000/d. d = 3000/tan(15°) ≈ 3000/0.2679 ≈ 11,197 ft.

8. Reference Angles

For an angle θ in standard position, the reference angle is the acute angle between the terminal side and the x-axis. Quadrant I: reference angle = θ. Quadrant II: reference angle = 180° - θ. Quadrant III: reference angle = θ - 180°. Quadrant IV: reference angle = 360° - θ. Example 1: θ = 150°. Reference angle = 180° - 150° = 30°. Example 2: θ = 225°. Reference angle = 225° - 180° = 45°. Example 3: θ = 310°. Reference angle = 360° - 310° = 50°.

9. CAST (Signs by Quadrant)

All: all trig functions are positive in Quadrant I. Sine: only sin (and csc) are positive in Quadrant II. Tangent: only tan (and cot) are positive in Quadrant III. Cosine: only cos (and sec) are positive in Quadrant IV. Mnemonic: All Students Take Calculus. Example: sin(120°). Reference angle = 60°. Quadrant II (sine positive). sin(120°) = sin(60°) = sqrt(3)/2. Example: cos(210°). Reference angle = 30°. Quadrant III (cosine negative). cos(210°) = -cos(30°) = -sqrt(3)/2.

10. Using a Calculator for Trig

Make sure your calculator is in DEGREE mode for degree problems (not radians). Example 1: Find sin(52°). In degree mode: sin(52°) ≈ 0.7880. Example 2: Find the angle θ if cos(θ) = 0.6. Use inverse: θ = arccos(0.6) ≈ 53.13°. Example 3: Find θ if tan(θ) = 2.5. θ = arctan(2.5) ≈ 68.2°. Inverse trig functions: arcsin, arccos, arctan (written sin^(-1), cos^(-1), tan^(-1) on calculators). Note: arcsin and arccos return angles in [0°, 90°]; arctan returns angles in (-90°, 90°).

11. Common Mistakes

- Using your calculator in radian mode when the problem is in degrees (or vice versa). - Mixing up opposite and adjacent when labeling the triangle. - Forgetting that tan(45°) = 1, not sqrt(2)/2. - Taking the inverse trig of a number outside [-1, 1] for sin or cos (impossible). - Not labeling which angle is θ before writing the trig ratios.