Common Trigonometric Function Values

Why This Matters

Trigonometric function values aren't just numbers to memorize—they're the foundation for nearly everything you'll encounter in calculus, physics, and engineering. You're being tested on your ability to recognize patterns in the unit circle, connect these values to special right triangles, and apply them fluently in equations, identities, and graphing problems. When you see a trig expression on an exam, you need instant recall—there's no time to derive these from scratch.

The good news? These values follow logical patterns rooted in geometry. The unit circle, 30-60-90 triangles, and 45-45-90 triangles generate every value you need. Once you understand why $\sin(30°) = \frac{1}{2}$ and how it relates to $\cos(60°)$, you'll stop seeing these as random facts and start recognizing the elegant symmetry underneath. Don't just memorize the table—know which triangle or circle property each value comes from.

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The Boundary Angles: 0° and 90°

These angles sit on the axes of the unit circle, making their values either 0 or 1. The coordinates of points on the axes have one component at its maximum and the other at zero.

sin(0°) = 0

• Unit circle position—at 0°, the point is $(1, 0)$, so the y-coordinate (sine) is zero
• Physical meaning: no vertical displacement; the angle lies flat along the positive x-axis
• Pattern recognition: sine starts at 0, increases to 1 at 90°, then decreases—this is your starting point for the sine wave

sin(90°) = 1

• Maximum value—at 90°, the point is $(0, 1)$, representing the highest point on the unit circle
• Range boundary: this is the upper limit of sine's range $[-1, 1]$
• Wave behavior: marks the peak of the sine function's first cycle

cos(0°) = 1

• Unit circle position—at 0°, the point is $(1, 0)$, so the x-coordinate (cosine) is one
• Maximum value: cosine begins at its peak and decreases toward 90°
• Complementary relationship: $\cos(0°) = \sin(90°)$ demonstrates the cofunction identity

cos(90°) = 0

• Unit circle position—at 90°, the point is $(0, 1)$, so the x-coordinate (cosine) is zero
• Tangent undefined: since $\tan(90°) = \frac{\sin(90°)}{\cos(90°)} = \frac{1}{0}$, tangent has a vertical asymptote here
• Pattern recognition: cosine hits zero at 90° and 270°—these are your asymptote locations for tangent and secant

tan(0°) = 0

• Ratio calculation—$\tan(0°) = \frac{\sin(0°)}{\cos(0°)} = \frac{0}{1} = 0$
• Slope interpretation: tangent represents the slope of the terminal side; a horizontal line has slope zero
• Graph behavior: the tangent function passes through the origin with a positive slope

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The 30-60-90 Triangle Values

These values come from the 30-60-90 special right triangle with side ratios $1 : \sqrt{3} : 2$. The short leg is opposite 30°, the long leg is opposite 60°, and the hypotenuse is 2.

sin(30°) = 1/2

• Triangle ratio—opposite side (1) divided by hypotenuse (2) gives $\frac{1}{2}$
• Memorization anchor: this is the only "nice" fraction among sine values—use it as your reference point
• Cofunction pair: $\sin(30°) = \cos(60°)$ because complementary angles swap sine and cosine

sin(60°) = √3/2

• Triangle ratio—opposite side ($\sqrt{3}$) divided by hypotenuse (2) gives $\frac{\sqrt{3}}{2}$
• Larger angle, larger sine: since 60° > 30°, its sine value is greater (in the first quadrant)
• Decimal approximation: $\approx 0.866$, useful for checking calculator work

cos(30°) = √3/2

• Triangle ratio—adjacent side ($\sqrt{3}$) divided by hypotenuse (2) gives $\frac{\sqrt{3}}{2}$
• Cofunction identity: $\cos(30°) = \sin(60°)$—same value, complementary angles
• Pattern: cosine of the smaller angle equals sine of the larger angle in a complementary pair

cos(60°) = 1/2

• Triangle ratio—adjacent side (1) divided by hypotenuse (2) gives $\frac{1}{2}$
• Symmetry with sine: $\cos(60°) = \sin(30°)$—the values swap for complementary angles
• Decreasing pattern: cosine decreases as angles increase from 0° to 90°

tan(30°) = √3/3

• Ratio calculation—$\tan(30°) = \frac{\sin(30°)}{\cos(30°)} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$
• Rationalized form: always rationalize denominators; $\frac{\sqrt{3}}{3}$ is preferred over $\frac{1}{\sqrt{3}}$
• Reciprocal relationship: $\tan(30°) = \frac{1}{\tan(60°)}$—tangent values of complementary angles are reciprocals

tan(60°) = √3

• Ratio calculation—$\tan(60°) = \frac{\sin(60°)}{\cos(60°)} = \frac{\sqrt{3}/2}{1/2} = \sqrt{3}$
• Slope interpretation: a 60° angle has a steep slope of approximately 1.732
• Reciprocal relationship: $\tan(60°) \cdot \tan(30°) = \sqrt{3} \cdot \frac{\sqrt{3}}{3} = 1$

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The 45-45-90 Triangle Values

These values come from the isosceles right triangle with side ratios $1 : 1 : \sqrt{2}$. Both legs are equal, creating perfect symmetry between sine and cosine.

sin(45°) = √2/2

• Triangle ratio—opposite side (1) divided by hypotenuse ($\sqrt{2}$) gives $\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$
• Equal to cosine: $\sin(45°) = \cos(45°)$ because 45° is its own complement
• Decimal approximation: $\approx 0.707$, exactly halfway between $\sin(30°)$ and $\sin(60°)$

cos(45°) = √2/2

• Triangle ratio—adjacent side (1) divided by hypotenuse ($\sqrt{2}$) gives $\frac{\sqrt{2}}{2}$
• Symmetry point: this is where sine and cosine curves intersect in the first quadrant
• Unit circle location: the point $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$ lies on the line $y = x$

tan(45°) = 1

• Ratio calculation—$\tan(45°) = \frac{\sin(45°)}{\cos(45°)} = \frac{\sqrt{2}/2}{\sqrt{2}/2} = 1$
• Slope interpretation: a 45° angle creates a line with slope 1 (rises 1 unit for every 1 unit run)
• Reference value: when solving $\tan(\theta) = 1$, your reference angle is always 45°

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Quick Reference Table

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Self-Check Questions

1. Which two trigonometric values both equal $\frac{\sqrt{3}}{2}$, and what geometric property explains why they're equal?

2. If you know $\sin(30°) = \frac{1}{2}$, how can you immediately determine $\cos(60°)$ without any calculation?

3. Compare $\tan(30°)$ and $\tan(60°)$: what is their product, and why does this relationship hold for any pair of complementary angles?

4. Why is $\tan(90°)$ undefined while $\tan(0°) = 0$? Explain using the ratio definition of tangent.

5. A problem asks you to evaluate $\sin^2(45°) + \cos^2(45°)$. Without calculating, what must the answer be, and what identity guarantees this result?