How to Calculate Trigonometric Functions: Sine, Cosine, and Tangent Using a Scientific Calculator

Trigonometric functions — sine (sin), cosine (cos), and tangent (tan) — are fundamental to geometry, physics, engineering, navigation, and even music theory. Whether you are calculating the height of a building using an angle of elevation, determining forces in a truss, or solving for angles in a right triangle, understanding how to use these functions on a calculator is essential. Use our free Scientific Calculator to compute any trigonometric function instantly.

Understanding Sine, Cosine, and Tangent

In a right triangle with angle θ:

Function	Formula	Pronounced	Range
sin(θ)	Opposite / Hypotenuse	Sign	−1 to 1
cos(θ)	Adjacent / Hypotenuse	Cosine	−1 to 1
tan(θ)	Opposite / Adjacent	Tangent	All real numbers

A handy mnemonic is SOH-CAH-TOA: Sine = Opposite over Hypotenuse, Cosine = Adjacent over Hypotenuse, Tangent = Opposite over Adjacent. Try these with the Scientific Calculator using any angle.

Degrees vs Radians: What You Need to Know

Scientific calculators operate in two angle modes:

Degrees (DEG): A full circle = 360°. Used in most everyday situations: navigation, construction, geography.
Radians (RAD): A full circle = 2π radians. Used in calculus, physics equations, and advanced math.

To convert: θ(rad) = θ(deg) × π / 180. For example, 45° = π/4 radians ≈ 0.7854 rad. The Scientific Calculator lets you toggle between modes so you always get the correct result.

Common Trigonometric Values

These angles appear frequently in problems and are worth memorizing:

Angle (°)	sin	cos	tan
0°	0	1	0
30°	1/2 (√3/3)	√3/2	1/√3 (≈ 0.577)
45°	√2/2 (≈ 0.707)	√2/2 (≈ 0.707)	1
60°	√3/2 (≈ 0.866)	1/2	√3 (≈ 1.732)
90°	1	0	Undefined

Verify these values using the Scientific Calculator to build confidence in using trig functions.

Real-World Applications of Trigonometry

Finding heights: Stand 50 ft from a building, measure the angle to the top (65°). Height = 50 × tan(65°) = 50 × 2.144 = 107 ft.
Navigation: Ships and aircraft use bearings (angles from north) to plot courses. Converting between bearing and Cartesian coordinates uses sin and cos.
Physics: Resolving forces into horizontal and vertical components. A 100 N force at 30° has horizontal = 100 × cos(30°) = 86.6 N and vertical = 100 × sin(30°) = 50 N.
Sound waves: Audio signals are modeled as sine waves. Frequency, amplitude, and phase are all described using trigonometric functions.

Use the Scientific Calculator at TodayCalculator.com for all your trigonometry calculations — whether in degrees or radians.

Inverse Trigonometric Functions

When you know the ratio but need the angle, use the inverse functions: sin⁻¹(x), cos⁻¹(x), tan⁻¹(x) (also written as arcsin, arccos, arctan). For example, if sin(θ) = 0.5, then θ = sin⁻¹(0.5) = 30°. The Scientific Calculator includes full inverse trig function support.