Essential Trigonometric Concepts

Why This Matters

Trigonometry shows up repeatedly on the PSAT, and here's the thing—you're not just being tested on whether you memorized SOH-CAH-TOA. The exam wants to know if you can apply these relationships to find missing sides, calculate angles, and solve real-world problems involving distances and heights. Questions often combine multiple concepts: you might need to recognize a special angle, apply the right ratio, and use the Pythagorean theorem all in one problem.

The good news? Trig on the PSAT is predictable. Master the core ratios, know your special angles cold, and understand how the unit circle connects everything together. Don't just memorize formulas—know when each tool applies and why the relationships work. That conceptual understanding is what separates students who guess from students who solve with confidence.

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The Foundation: Right Triangle Relationships

Every trig concept on the PSAT traces back to one simple idea: the ratios between sides of a right triangle stay constant for any given angle, regardless of the triangle's size.

Right Triangle Trigonometry

• SOH-CAH-TOA defines the three core ratios—these relate any angle to the triangle's opposite side, adjacent side, and hypotenuse
• The reference angle determines which side is "opposite" and which is "adjacent"—the hypotenuse is always across from the 90° angle
• Real-world applications dominate PSAT questions—think ladders against walls, shadows, and line-of-sight problems

Pythagorean Theorem and Its Relationship to Trigonometry

• $a^2 + b^2 = c^2$ connects all three sides—where $c$ is always the hypotenuse (longest side, opposite the right angle)
• Use this to find a missing side before applying trig ratios—you often need all three sides to check your work
• The theorem underlies the fundamental identity $\sin^2(\theta) + \cos^2(\theta) = 1$—this isn't coincidence; it's the Pythagorean theorem in disguise

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The Core Ratios: Sine, Cosine, and Tangent

These three functions are your primary tools. Each ratio compares two specific sides of a right triangle relative to a chosen angle.

Sine, Cosine, and Tangent Functions

• Sine: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$—use when you have or need the side across from your angle
• Cosine: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$—use when you have or need the side touching your angle
• Tangent: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$—use when the hypotenuse isn't involved at all

Inverse Trigonometric Functions

• Arcsin, arccos, and arctan "undo" the trig functions—they take a ratio and return an angle
• Use inverse functions when you know sides but need the angle—written as $\sin^{-1}$, $\cos^{-1}$, or $\tan^{-1}$ on your calculator
• Output is always an angle in degrees or radians—make sure your calculator is in the right mode before solving

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Special Angles and Memorized Values

Certain angles appear so frequently that the PSAT expects you to know their trig values instantly. These "special angles" come from two specific right triangles: the 45-45-90 and the 30-60-90.

Trigonometric Ratios and Special Angles

• The 45-45-90 triangle gives you $\sin(45°) = \cos(45°) = \frac{\sqrt{2}}{2}$—this isoceles right triangle has legs in ratio $1:1:\sqrt{2}$
• The 30-60-90 triangle provides values for both angles—legs in ratio $1:\sqrt{3}:2$, giving $\sin(30°) = \frac{1}{2}$ and $\sin(60°) = \frac{\sqrt{3}}{2}$
• Memorize the five key angles: 0°, 30°, 45°, 60°, 90°—these appear constantly and save precious time when you know them cold

Unit Circle

• A circle with radius 1 centered at the origin—every point on it has coordinates $(\cos(\theta), \sin(\theta))$
• Extends trig beyond right triangles to any angle—including angles greater than 90° or negative angles
• The x-coordinate is cosine, the y-coordinate is sine—this pattern holds for all four quadrants

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Solving Problems: Putting It All Together

Here's where concepts become points. Solving trig problems means choosing the right tool, setting up the equation correctly, and executing the algebra.

Solving Right Triangles

• "Solving" means finding all unknown sides and angles—you typically start with one angle (besides 90°) and one side
• Choose your ratio based on what you know and what you need—if you have the hypotenuse and need the opposite side, use sine
• Always check that your answers make sense—the hypotenuse must be the longest side, and angles must sum to 180°

Basic Trigonometric Identities

• The Pythagorean identity $\sin^2(\theta) + \cos^2(\theta) = 1$ simplifies expressions—use it to convert between sine and cosine
• The quotient identity $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ connects all three ratios—helpful when a problem gives you sine and cosine separately
• Reciprocal identities define cosecant, secant, and cotangent—these rarely appear on PSAT but know that $\csc(\theta) = \frac{1}{\sin(\theta)}$

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Angle Measurement and Visualization

Understanding how we measure angles and what trig functions look like helps you catch errors and interpret results.

Radian Measure

• Radians measure angles using arc length—one radian is the angle where the arc length equals the radius
• Key conversion: $180° = \pi$ radians—so $90° = \frac{\pi}{2}$, $60° = \frac{\pi}{3}$, $45° = \frac{\pi}{4}$, $30° = \frac{\pi}{6}$
• PSAT may use either unit—check the answer choices to see which format they expect

Graphing Trigonometric Functions

• Sine and cosine waves oscillate between -1 and 1—this range is called the amplitude
• The period is how long until the pattern repeats—for basic sine and cosine, the period is $2\pi$ radians or 360°
• Phase shift moves the graph left or right; vertical shift moves it up or down—these transformations appear in equations like $y = \sin(x - \frac{\pi}{2}) + 3$

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Quick Reference Table

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Self-Check Questions

1. You know one acute angle and the hypotenuse of a right triangle. Which trig function would you use to find the side opposite that angle, and which would you use for the side adjacent to it?

2. Without a calculator, what are $\sin(45°)$ and $\cos(30°)$? Which special triangles do these values come from?

3. Compare and contrast: When would you use the Pythagorean theorem versus a trig ratio to find a missing side? Give an example situation for each.

4. A problem gives you $\sin(\theta) = 0.6$ and asks for the angle. What function do you use, and what should you check on your calculator before solving?

5. If a point on the unit circle has coordinates $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$, what angle does this represent, and how do you know?