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Dec 9, 2025

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2 pages

Understanding Trigonometry: Solving Angles in Degrees and Radians

Kaylin Sheffer

@kaylinshef

Trigonometry builds on the relationship between angles and sides of... Show more

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Unit Circle & Quadrants

The unit circle helps us understand trig functions across all four quadrants. In this circle with radius 1, each point $(x,y)$ corresponds to specific trig values. Remember that $x$ represents the adjacent side (cosine) and $y$ represents the opposite side (sine).

Each quadrant has its own sign pattern for trig functions. Sine is positive in Q1 and Q2, while cosine is positive in Q1 and Q4. Tangent is positive in Q1 and Q3. A helpful memory trick is "All Students Take Calculus" - All (Q1), Sine (Q2), Tangent (Q3), Cosine (Q4).

To find missing coordinates on the unit circle, use the Pythagorean identity: $x^2+y^2=1$ . For example, if you know a point has $y=\frac{4}{7}$ , you can find $x$ by solving $x^2+(\frac{4}{7})^2=1$ , which gives $x=\pm\frac{\sqrt{33}}{7}$ .

💡 When solving trig problems, always identify the quadrant first! This tells you whether your final answer should be positive or negative.

Radians & Solving for θ

Radians measure angles using the arc length created on a unit circle. One complete revolution equals $2\pi$ radians or $360°$ . Common angles include $\frac{\pi}{6}$ (30°), $\frac{\pi}{4}$ (45°), and $\frac{\pi}{3}$ (60°).

When solving for an angle θ, you need to find all possible solutions within your given range. For example, if $\sin θ = \frac{-\sqrt{3}}{2}$ and θ is in the range $[0°, 360°]$ , you must determine which quadrants have negative sine (Q3 and Q4) and find the reference angles.

Converting between degrees and radians is straightforward - multiply degrees by $\frac{\pi}{180°}$ . For example, $210° = 30° × 7 = \frac{\pi}{6} × 7 = \frac{7\pi}{6}$ . When dealing with large radian values like $13\frac{\pi}{4}$ , subtract multiples of $2\pi$ to find the equivalent position on the unit circle.

💡 When solving for θ with a negative trig value, always check both possible quadrants where that function is negative!

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Algebra 2

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Dec 9, 2025

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2 pages

Understanding Trigonometry: Solving Angles in Degrees and Radians

Kaylin Sheffer

@kaylinshef

Trigonometry builds on the relationship between angles and sides of triangles. Day 2 of basic trig covers the unit circle, quadrants, and solving for angles in both degrees and radians - concepts you'll need for everything from calculus to engineering.

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Unit Circle & Quadrants

💡 When solving trig problems, always identify the quadrant first! This tells you whether your final answer should be positive or negative.

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Radians & Solving for θ

💡 When solving for θ with a negative trig value, always check both possible quadrants where that function is negative!