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Mom's Basement University

Trigonometry Review

What You’ll Learn

Section titled “What You’ll Learn”

This is the capstone review for the Trigonometry section. It brings together all the major concepts you’ve learned, with mixed practice, key formulas, and real-world connections to help everything come together.

Major Topics Covered

Section titled “Major Topics Covered”

1. Right Triangle Trigonometry

Section titled “1. Right Triangle Trigonometry”
  • SOH CAH TOA: the three fundamental ratios
  • sin θ = opposite / hypotenuse
  • cos θ = adjacent / hypotenuse
  • tan θ = opposite / adjacent
  • Solving right triangles for missing sides and angles
  • Applications: angles of elevation/depression, ladders, ramps

2. Radians and the Unit Circle

Section titled “2. Radians and the Unit Circle”
  • Full circle = 360° = 2π radians
  • To convert: degrees × π/180, or radians × 180/π
  • Key angles: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2
  • The unit circle: radius 1, any point is (cos θ, sin θ)
  • ASTC: All, Sine, Tangent, Cosine (signs by quadrant)

3. Graphing Trig Functions

Section titled “3. Graphing Trig Functions”
  • Sine and cosine: amplitude, period (2π), midline, phase shift
  • General form: y = a sin(b(x - c)) + d
  • Tangent: period π, vertical asymptotes where cos θ = 0
  • Reciprocal functions: csc, sec, cot
  • Transformations: amplitude = |a|, period = 2π/b, phase shift = c, vertical shift = d

4. Trigonometric Identities

Section titled “4. Trigonometric Identities”
  • Pythagorean: sin²θ + cos²θ = 1
  • Reciprocal: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ
  • Quotient: tan θ = sin θ / cos θ
  • Sum and difference: sin(A ± B), cos(A ± B), tan(A ± B)
  • Verifying identities: work one side to match the other

5. Solving Trig Equations

Section titled “5. Solving Trig Equations”
  • Isolate the trig function, then use inverse trig
  • Account for all solutions in [0, 2π)
  • For multiple angles: solve for the inner expression first, then divide

6. Law of Sines and Law of Cosines

Section titled “6. Law of Sines and Law of Cosines”
  • Law of Sines: a/sin A = b/sin B = c/sin C (use for AAS, ASA, SSA)
  • Law of Cosines: a² = b² + c² - 2bc cos A (use for SAS, SSS)
  • SSA can produce 0, 1, or 2 triangles (ambiguous case)
  • Area formula: Area = ½ ab sin C

Mixed Worked Examples

Section titled “Mixed Worked Examples”

1. Right Triangle Application

A 25-foot ladder makes a 68° angle with the ground. How high up the wall does it reach?

The height is opposite the 68° angle, and the ladder is the hypotenuse. That’s SOH: sin = opposite / hypotenuse.

h=25×sin⁡(68°)≈25×0.9272≈23.18 feeth = 25 \times \sin(68°) \approx 25 \times 0.9272 \approx 23.18 \text{ feet}

2. Unit Circle

Find sin(2π/3) and cos(2π/3).

2π/3 = 120°, which is in Quadrant II. The reference angle is 60°. In Q2, sine is positive and cosine is negative:

sin⁡(2π3)=32,cos⁡(2π3)=−12\sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2}, \quad \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}

3. Graph Transformation

For y = 3 sin(2(x - π/4)) + 1, identify the key features:

Amplitude=3,Period=2π2=π,Phase shift=π4 right,Vertical shift=+1\text{Amplitude} = 3, \quad \text{Period} = \frac{2\pi}{2} = \pi, \quad \text{Phase shift} = \frac{\pi}{4} \text{ right}, \quad \text{Vertical shift} = +1

4. Sum Identity

Find sin(75°) exactly.

sin⁡(75°)=sin⁡(45°+30°)=sin⁡45°cos⁡30°+cos⁡45°sin⁡30°\sin(75°) = \sin(45° + 30°) = \sin 45° \cos 30° + \cos 45° \sin 30° =22⋅32+22⋅12=6+24= \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4}

5. Law of Sines

In triangle ABC, angle A = 40°, angle B = 65°, side a = 12. Find side b.

First, angle C = 180° - 40° - 65° = 75°. Then:

b=12×sin⁡65°sin⁡40°≈12×0.90630.6428≈16.92 cmb = \frac{12 \times \sin 65°}{\sin 40°} \approx \frac{12 \times 0.9063}{0.6428} \approx 16.92 \text{ cm}

Real-World Connections

Section titled “Real-World Connections”

Trigonometry helps you understand and model:

  • Waves (sound, light, ocean tides)
  • Rotational motion and circular paths
  • Navigation and surveying
  • Construction and engineering (ramps, roofs, bridges)
  • Physics (projectile motion, forces at angles)
  • Music and audio (sound waves are sine waves)
  • Computer graphics and game development (rotation, lighting, 3D rendering)
  • Medical imaging (CT scans use trigonometric reconstruction)

Final Tips for Success

Section titled “Final Tips for Success”
  • Draw clear diagrams for triangle problems
  • Memorize the unit circle values for the major angles
  • Use SOH CAH TOA for right triangles and Law of Sines/Cosines for general triangles
  • Check solutions in the original equation (especially when squaring or using identities)
  • Practice converting between degrees and radians
  • Sketch graphs when solving or verifying

Comprehensive Review Quiz

Section titled “Comprehensive Review Quiz”
In a right triangle, $\sin\theta$ equals:
Convert 135° to radians:
On the unit circle, $\cos(\pi/3)$ equals:
The period of $y = \sin(x)$ is:
The amplitude of $y = 4\sin(3x) - 2$ is:
The identity $\sin^2\theta + \cos^2\theta = 1$ is a:
Solve $\sin(2\theta) = \frac{1}{2}$ for $\theta$ in $[0, 2\pi)$. The solutions are:
The Law of Sines is most useful when you know:
In $y = 2\cos(4(x - \pi/6)) + 1$, the phase shift is:
The Law of Cosines is best used when you know:
The period of $y = \tan(x)$ is:
What is $\cos(5\pi/6)$?
When solving trig equations, you should always:
The reciprocal of sine is:
A ramp rises 5 feet over 60 feet horizontally. The angle is approximately:
The sum identity for cosine is:
In the ambiguous case (SSA), the number of possible triangles can be:
The period of $y = \cos(2x)$ is:
Trigonometry is most directly useful for solving problems involving:
The most important final step when solving any trig equation is: