Visual Tools
Calculators
Tables
Mathematical Keyboard
Converters
Other Tools

Trigonometry

Trigonometry Formulas

Introduction to Trigonometry

While mainly focusing on relationships between the angles and sides of triangles, trigonometry extends beyond simple geometric shapes, offering tools to describe patterns involving periodicity and circular motion. This makes trigonometry essential in fields ranging from physics and engineering to computer graphics and navigation.

Here’s what students can expect to learn:

1. Core Concepts:
• Angles and Their Measures: Degrees, radians, and how they relate to circles.
• Trigonometric Functions: Sine, cosine, tangent, and their reciprocals (cosecant, secant, cotangent).
• Unit Circle: A key tool for understanding how trigonometric functions behave across all angles.
• Inverse Trigonometric Functions: Solving equations to find angles.

2. Applications:
• Solving Triangles: Using the laws of sines and cosines to find unknown sides or angles.
• Wave Behavior: Modeling oscillations in physics, sound waves, and light waves.
• Circular and Periodic Motion: Understanding orbits, pendulums, and cycles.

3. Advanced Topics:
• Identities: Simplifying expressions and solving equations using formulas like the Pythagorean identity, angle addition formulas, and double-angle formulas.
• Graphs of Trigonometric Functions: Visualizing periodic patterns and their transformations (shifts, stretches, reflections).
• Complex Numbers and Euler’s Formula: Exploring the deep connection between trigonometry and algebra.

4. Skills Developed:
• Problem-solving: Tackling real-world problems involving geometry, motion, and oscillation.
• Visualization: Interpreting and sketching graphs and diagrams to represent abstract relationships.
• Logical reasoning: Manipulating trigonometric expressions and proofs.

Trigonometry teaches students how to connect abstract mathematical concepts with practical scenarios, developing versatile skills useful in scientific modeling, technical fields, and everyday problem-solving.

Trigonometry Formulas

Explore Trigonometry formulas with explanations and examples

Sine Function (sin)

sin⁡θ=Opposite SideHypotenuse\sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}}

Cosine Function (cos)

cos⁡θ=Adjacent SideHypotenuse\cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}

Tangent Function (tan)

tan⁡θ=Opposite SideAdjacent Side\tan \theta = \frac{\text{Opposite Side}}{\text{Adjacent Side}}

Cosecant Function (csc)

csc⁡θ=HypotenuseOpposite Side=1sin⁡θ\csc \theta = \frac{\text{Hypotenuse}}{\text{Opposite Side}} = \frac{1}{\sin \theta}

Secant Function (sec)

sec⁡θ=HypotenuseAdjacent Side=1cos⁡θ\sec \theta = \frac{\text{Hypotenuse}}{\text{Adjacent Side}} = \frac{1}{\cos \theta}

Cotangent Function (cot)

cot⁡θ=Adjacent SideOpposite Side=1tan⁡θ\cot \theta = \frac{\text{Adjacent Side}}{\text{Opposite Side}} = \frac{1}{\tan \theta}

Secant Reciprocal Identity

sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}

Cosecant Reciprocal Identity

csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}

Cotangent Reciprocal Identity

cot⁡θ=1tan⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}

Primary Pythagorean Identity

sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1

Tangent Pythagorean Identity

1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta

Cotangent Pythagorean Identity

1+cot⁡2θ=csc⁡2θ1 + \cot^2 \theta = \csc^2 \theta

Sine-Cosine Co-Function Identity

sin⁡(π2−θ)=cos⁡θ\sin\left(\frac{\pi}{2} - \theta\right) = \cos \theta

Cosine-Sine Co-Function Identity

cos⁡(π2−θ)=sin⁡θ\cos\left(\frac{\pi}{2} - \theta\right) = \sin \theta

Tangent-Cotangent Co-Function Identity

tan⁡(π2−θ)=cot⁡θ\tan\left(\frac{\pi}{2} - \theta\right) = \cot \theta

Cotangent-Tangent Co-Function Identity

cot⁡(π2−θ)=tan⁡θ\cot\left(\frac{\pi}{2} - \theta\right) = \tan \theta

Secant-Cosecant Co-Function Identity

sec⁡(π2−θ)=csc⁡θ\sec\left(\frac{\pi}{2} - \theta\right) = \csc \theta

Cosecant-Secant Co-Function Identity

csc⁡(π2−θ)=sec⁡θ\csc\left(\frac{\pi}{2} - \theta\right) = \sec \theta

Cosine Even Identity

cos⁡(−θ)=cos⁡θ\cos(-\theta) = \cos \theta

Secant Even Identity

sec⁡(−θ)=sec⁡θ\sec(-\theta) = \sec \theta

Sine Odd Identity

sin⁡(−θ)=−sin⁡θ\sin(-\theta) = -\sin \theta

Tangent Odd Identity

tan⁡(−θ)=−tan⁡θ\tan(-\theta) = -\tan \theta

Cosecant Odd Identity

csc⁡(−θ)=−csc⁡θ\csc(-\theta) = -\csc \theta

Cotangent Odd Identity

cot⁡(−θ)=−cot⁡θ\cot(-\theta) = -\cot \theta

Sine Function (sin)

sin⁡θ=Opposite SideHypotenuse\sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}}

Cosine Function (cos)

cos⁡θ=Adjacent SideHypotenuse\cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}

Tangent Function (tan)

tan⁡θ=Opposite SideAdjacent Side\tan \theta = \frac{\text{Opposite Side}}{\text{Adjacent Side}}

Cosecant Function (csc)

csc⁡θ=HypotenuseOpposite Side=1sin⁡θ\csc \theta = \frac{\text{Hypotenuse}}{\text{Opposite Side}} = \frac{1}{\sin \theta}

Secant Function (sec)

sec⁡θ=HypotenuseAdjacent Side=1cos⁡θ\sec \theta = \frac{\text{Hypotenuse}}{\text{Adjacent Side}} = \frac{1}{\cos \theta}

Cotangent Function (cot)

cot⁡θ=Adjacent SideOpposite Side=1tan⁡θ\cot \theta = \frac{\text{Adjacent Side}}{\text{Opposite Side}} = \frac{1}{\tan \theta}

Secant Reciprocal Identity

sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}

Cosecant Reciprocal Identity

csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}

Cotangent Reciprocal Identity

cot⁡θ=1tan⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}

Primary Pythagorean Identity

sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1

Tangent Pythagorean Identity

1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta

Cotangent Pythagorean Identity

1+cot⁡2θ=csc⁡2θ1 + \cot^2 \theta = \csc^2 \theta

Sine-Cosine Co-Function Identity

sin⁡(π2−θ)=cos⁡θ\sin\left(\frac{\pi}{2} - \theta\right) = \cos \theta

Cosine-Sine Co-Function Identity

cos⁡(π2−θ)=sin⁡θ\cos\left(\frac{\pi}{2} - \theta\right) = \sin \theta

Tangent-Cotangent Co-Function Identity

tan⁡(π2−θ)=cot⁡θ\tan\left(\frac{\pi}{2} - \theta\right) = \cot \theta

Cotangent-Tangent Co-Function Identity

cot⁡(π2−θ)=tan⁡θ\cot\left(\frac{\pi}{2} - \theta\right) = \tan \theta

Secant-Cosecant Co-Function Identity

sec⁡(π2−θ)=csc⁡θ\sec\left(\frac{\pi}{2} - \theta\right) = \csc \theta

Cosecant-Secant Co-Function Identity

csc⁡(π2−θ)=sec⁡θ\csc\left(\frac{\pi}{2} - \theta\right) = \sec \theta

Cosine Even Identity

cos⁡(−θ)=cos⁡θ\cos(-\theta) = \cos \theta

Secant Even Identity

sec⁡(−θ)=sec⁡θ\sec(-\theta) = \sec \theta

Sine Odd Identity

sin⁡(−θ)=−sin⁡θ\sin(-\theta) = -\sin \theta

Tangent Odd Identity

tan⁡(−θ)=−tan⁡θ\tan(-\theta) = -\tan \theta

Cosecant Odd Identity

csc⁡(−θ)=−csc⁡θ\csc(-\theta) = -\csc \theta

Cotangent Odd Identity

cot⁡(−θ)=−cot⁡θ\cot(-\theta) = -\cot \theta