Trigonometry Formula - Complete Guide for Students and Learners

Trigonometry, a crucial branch of mathematics, finds its application in geometry, physics, engineering, and even in daily problem-solving. At the heart of trigonometry lies a powerful set of equations known as trigonometry formulas. Whether you are a student preparing for exams or a professional brushing up on the basics, having a solid grasp of these formulas can make all the difference.

In this article, we will cover over 1500+ words of detailed yet simplified explanations of every essential trigonometry formula. From basic identities to advanced formulas used in real-world applications, this guide is your one-stop reference.

What is Trigonometry?

Trigonometry is the study of relationships between angles and sides of triangles. It primarily focuses on right-angled triangles but extends to general triangles and unit circles. The most fundamental components of trigonometry are the six trigonometric functions:

Sine (sin)
Cosine (cos)
Tangent (tan)
Cosecant (csc)
Secant (sec)
Cotangent (cot)

Each of these functions describes a ratio derived from the angles and sides of a triangle.

Why Trigonometry Formulas Matter

Trigonometry formulas simplify complex geometric problems and help solve:

Heights and distances
Wave and oscillation equations
Electrical engineering problems
Architecture and design
Astronomy and satellite navigation

For academic purposes, mastering these formulas is essential for scoring well in mathematics, physics, and engineering exams.

List of Basic Trigonometry Formulas

1. Trigonometric Ratios (For a right-angled triangle):

Let the triangle have an angle $θ\theta$ and the sides: Opposite (O), Adjacent (A), and Hypotenuse (H).

$sin⁡θ=OH\sin \theta = \frac{O}{H}$
$cos⁡θ=AH\cos \theta = \frac{A}{H}$
$tan⁡θ=OA\tan \theta = \frac{O}{A}$
$csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}$
$sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}$
$cot⁡θ=1tan⁡θ\cot \theta = \frac{1}{\tan \theta}$

2. Reciprocal Identities

$csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}$
$sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}$
$cot⁡θ=1tan⁡θ\cot \theta = \frac{1}{\tan \theta}$

3. Pythagorean Identities

$sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1$
$1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta$
$1+cot⁡2θ=csc⁡2θ1 + \cot^2 \theta = \csc^2 \theta$

Angle Conversion Formulas

Convert between degrees and radians:

Degrees to Radians: $θrad=θdeg×π180\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}$
Radians to Degrees: $θdeg=θrad×180π\theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180}{\pi}$

Trigonometry Formulas for Allied Angles

These formulas help in simplifying expressions involving $90∘,180∘,270∘,90^\circ, 180^\circ, 270^\circ,$ and $360∘360^\circ$ :

$sin⁡(180∘−θ)=sin⁡θ\sin(180^\circ – \theta) = \sin \theta$
$cos⁡(180∘−θ)=−cos⁡θ\cos(180^\circ – \theta) = -\cos \theta$
$tan⁡(180∘−θ)=−tan⁡θ\tan(180^\circ – \theta) = -\tan \theta$

Trigonometric Values Table

Common angles and their trigonometric values:

Angle	sin $θ\theta$	cos $θ\theta$	tan $θ\theta$
0°	0	1	0
30°	1/2	$3/2\sqrt{3}/2$	1/ $3\sqrt{3}$
45°	$2/2\sqrt{2}/2$	$2/2\sqrt{2}/2$	1
60°	$3/2\sqrt{3}/2$	1/2	$3\sqrt{3}$
90°	1	0	Undefined

Sum and Difference Formulas

$sin⁡(a±b)=sin⁡acos⁡b±cos⁡asin⁡b\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b$
$cos⁡(a±b)=cos⁡acos⁡b∓sin⁡asin⁡b\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b$
$tan⁡(a±b)=tan⁡a±tan⁡b1∓tan⁡atan⁡b\tan(a \pm b) = \frac{\tan a \pm \tan b}{1 \mp \tan a \tan b}$

Double Angle and Half Angle Formulas

Double Angle:

$sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2\sin \theta \cos \theta$
$cos⁡2θ=cos⁡2θ−sin⁡2θ=2cos⁡2θ−1=1−2sin⁡2θ\cos 2\theta = \cos^2 \theta – \sin^2 \theta = 2\cos^2 \theta – 1 = 1 – 2\sin^2 \theta$
$tan⁡2θ=2tan⁡θ1−tan⁡2θ\tan 2\theta = \frac{2\tan \theta}{1 – \tan^2 \theta}$

Half Angle:

$sin⁡θ2=±1−cos⁡θ2\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 – \cos \theta}{2}}$
$cos⁡θ2=±1+cos⁡θ2\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}$
$tan⁡θ2=±1−cos⁡θ1+cos⁡θ\tan \frac{\theta}{2} = \pm \sqrt{\frac{1 – \cos \theta}{1 + \cos \theta}}$

Product to Sum Formulas

$sin⁡Asin⁡B=12[cos⁡(A−B)−cos⁡(A+B)]\sin A \sin B = \frac{1}{2}[\cos(A – B) – \cos(A + B)]$
$cos⁡Acos⁡B=12[cos⁡(A−B)+cos⁡(A+B)]\cos A \cos B = \frac{1}{2}[\cos(A – B) + \cos(A + B)]$
$sin⁡Acos⁡B=12[sin⁡(A+B)+sin⁡(A−B)]\sin A \cos B = \frac{1}{2}[\sin(A + B) + \sin(A – B)]$

Inverse Trigonometry Formulas

$sin⁡−1(sin⁡x)=x\sin^{-1}(\sin x) = x$ when $−π2≤x≤π2-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$
$cos⁡−1(cos⁡x)=x\cos^{-1}(\cos x) = x$ when $0≤x≤π0 \leq x \leq \pi$
$tan⁡−1(tan⁡x)=x\tan^{-1}(\tan x) = x$ when $−π2<x<π2-\frac{\pi}{2} < x < \frac{\pi}{2}$

Real-World Applications of Trigonometry Formulas

Engineering: Calculating forces, waveforms, and structural angles.
Physics: Analyzing projectile motion and light angles.
Architecture: Creating stable and aesthetic designs.
Astronomy: Measuring distances between celestial bodies.
GPS and Navigation: Locating coordinates and paths.

Tips to Remember Trigonometry Formulas

Use mnemonics like SOH-CAH-TOA.
Practice unit circle memorization.
Visual learning through triangles and graphs.
Regular revision and formula sheets.

Final Thoughts

Understanding and memorising trigonometry formulas is foundational for academic excellence and real-world applications. This comprehensive guide helps you learn the essential formulas with clarity, precision, and context.

Whether you’re revising for an exam or enhancing your mathematical skills, this SEO-friendly article on trigonometry formula is your reliable learning companion.

Stay tuned for downloadable PDF formula sheets, quizzes, and more educational content coming soon!