Trigonometry
sin, cos, tan + inverse
About This Calculator
Trigonometry studies the relationships between the angles and sides of triangles. The three primary functions — sine, cosine, and tangent — relate an angle in a right triangle to ratios of its sides. These functions and their inverses are fundamental to physics, engineering, navigation, and signal processing.
Formula
sin(θ) = opposite / hypotenuse
cos(θ) = adjacent / hypotenuse
tan(θ) = opposite / adjacent = sin(θ) / cos(θ)
Pythagorean identity: sin²(θ) + cos²(θ) = 1
Example Calculation
Find sin, cos, and tan for θ = 30°.
- sin(30°) = 0.5
- cos(30°) = sqrt(3)/2 ≈ 0.866
- tan(30°) = sin/cos = 0.5/0.866 ≈ 0.577
sin(30°)=0.5, cos(30°)=0.866, tan(30°)=0.577
Exact Trig Values for Common Angles
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 0.5 | 0.866 | 0.577 |
| 45° | 0.707 | 0.707 | 1 |
| 60° | 0.866 | 0.5 | 1.732 |
| 90° | 1 | 0 | undefined |
Frequently Asked Questions
What is the difference between degrees and radians?
Degrees divide a full circle into 360 parts. Radians measure angles by arc length on a unit circle; a full circle = 2π radians. Convert: radians = degrees × (π/180).
What are inverse trig functions used for?
Inverse functions (arcsin, arccos, arctan) find the angle when you know a ratio. For example, if sin(θ) = 0.5, then θ = arcsin(0.5) = 30°.
What does SOH-CAH-TOA mean?
It is a memory aid: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent. It applies to right triangles only.
Why is tan(90°) undefined?
tan(90°) = sin(90°)/cos(90°) = 1/0, which is undefined. As θ approaches 90°, tan(θ) increases without bound toward positive or negative infinity.