Sine, Cosine, and Tangent
Understand the three primary trig functions — what they measure, how to read them from a right triangle, and how to evaluate them at key angles.
Sine, cosine, and tangent are the core functions of trigonometry — and they appear everywhere in science, engineering, and technology. Signal processing, audio waveforms, AC electricity, GPS positioning, and structural analysis all rely on these three functions. They are the mathematical language of waves, oscillations, and circular motion. Understanding what sin, cos, and tan measure is the foundation for every trig application that follows.
- Define sine, cosine, and tangent as ratios of sides in a right triangle
- Evaluate sin, cos, and tan at the key angles 0°, 30°, 45°, 60°, 90°
- Use the reciprocal functions: cosecant, secant, and cotangent
These ratios depend only on the angle — not the size of the triangle. Any right triangle with a 30° angle will always have the same sin(30°) = 0.5, regardless of how large or small the triangle is.
Key Angle Values — Sin, Cos, Tan
| Angle | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | \(\frac{1}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{\sqrt{3}}\) |
| 45° | \(\frac{\sqrt{2}}{2}\) | \(\frac{\sqrt{2}}{2}\) | 1 |
| 60° | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{2}\) | \(\sqrt{3}\) |
| 90° | 1 | 0 | undefined |
Memory trick for sin: 0, ½, √2/2, √3/2, 1 — cos is sin in reverse order.
Find sin(60°), cos(60°), and tan(60°) exactly.
\[ \sin 60° = \frac{\sqrt{3}}{2} \approx 0.866 \qquad \cos 60° = \frac{1}{2} = 0.5 \qquad \tan 60° = \sqrt{3} \approx 1.732 \]In a right triangle with θ = 35°, hypotenuse = 20, what are the opposite and adjacent sides?
- Opposite: \(\text{opp} = 20 \sin 35° = 20 \times 0.574 \approx 11.47\)
- Adjacent: \(\text{adj} = 20 \cos 35° = 20 \times 0.819 \approx 16.38\)
- Check: \(\tan 35° = \frac{11.47}{16.38} \approx 0.700\) ✓ (matches table value)
A 24-foot rafter makes a 40° angle with the horizontal. What is the vertical rise of the roof?
\[ \text{rise} = 24 \times \sin 40° = 24 \times 0.643 \approx 15.4 \text{ ft} \]The roof rises about 15.4 feet. Sine gives the vertical component — the "opposite" side — when the hypotenuse (rafter) and angle are known.
- What is \(\sin 30° + \cos 60°\)?
- In a right triangle with hypotenuse 15 and angle 50°, find the opposite and adjacent sides.
- Why is \(\tan 90°\) undefined?
▶ Show Answers
- \(\frac{1}{2} + \frac{1}{2} =\) 1.
- opp = \(15\sin50° \approx 11.49\); adj = \(15\cos50° \approx 9.64\).
- \(\tan 90° = \frac{\sin90°}{\cos90°} = \frac{1}{0}\) — division by zero is undefined. The adjacent side has length zero at 90°.
- "Opposite" and "adjacent" change with the angle: Which side is opposite and which is adjacent depends on which angle θ you're working with. Label them relative to the angle — not the triangle's orientation.
- Confusing sin and cos values at 30° and 60°: sin(30°) = ½ = cos(60°); sin(60°) = √3/2 = cos(30°). They swap. Remember: sin increases from 0 to 1 as angle grows from 0° to 90°; cos decreases.
- Forgetting the Pythagorean identity: \(\sin^2\theta + \cos^2\theta = 1\) — always. Use this to find one value if you know the other.
- sin = opp/hyp, cos = adj/hyp, tan = opp/adj — memorize these three ratios.
- Key values at 0°, 30°, 45°, 60°, 90° should be recognizable on sight.
- \(\sin^2\theta + \cos^2\theta = 1\) — the Pythagorean identity, true for any angle.
- tan θ = sin θ / cos θ — tangent is always the ratio of the other two.
AC electricity is described by sine waves: \(V(t) = V_{peak} \sin(2\pi f t)\). Electrical engineers use sin and cos constantly to analyze phase relationships, power factors, and signal components. The "real" and "imaginary" parts of a phasor are the cosine and sine components. In signal processing, the Fourier transform decomposes any signal into sine and cosine waves — making these two functions the foundation of all digital audio, radio, and communications.
Calculator Connection
The Trigonometric Functions Calculator evaluates sin, cos, tan (and their reciprocals) for any angle in degrees or radians. The Right Triangle Solver finds all sides and angles of a right triangle from any two known values.
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Sine, Cosine, and Tangent — Quiz
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