Trigonometry Basics — Sin, Cos, Tan and Essential Identities Sin, cos, and tan aren't just three hard-to-remember letters — they're tools for solving a wide range of problems involving triangles, angles, and real-world applications. Learn them once, use them forever.
MathPal Team
February 28, 2026
3 min read Trigonometry is a topic many students "learn and forget," mostly because they memorize tables of values without understanding the underlying concepts. This post helps you understand it correctly — and remember it for good.
What is trigonometry?
Trigonometry studies the relationship between the sides and angles of triangles . It starts with right triangles, then extends to the unit circle and all real numbers.
Definitions in a right triangle
For a right triangle with acute angle α \alpha α a a a b b b c c c
sin α = a c cos α = b c tan α = a b \sin\alpha = \frac{a}{c} \qquad \cos\alpha = \frac{b}{c} \qquad \tan\alpha = \frac{a}{b} sin α = c a cos α = c b tan α = b a
cot α = b a = 1 tan α \cot\alpha = \frac{b}{a} = \frac{1}{\tan\alpha} cot α = a b = t a n α 1
Memory trick — SOH-CAH-TOA:
S in = O pposite / H ypotenuseC os = A djacent / H ypotenuseT an = O pposite / A djacent
Values at special angles
Angle 0 ° 0° 0° 30 ° 30° 30° 45 ° 45° 45° 60 ° 60° 60° 90 ° 90° 90° sin \sin sin 0 0 0 1 2 \dfrac{1}{2} 2 1 2 2 \dfrac{\sqrt{2}}{2} 2 2 3 2 \dfrac{\sqrt{3}}{2} 2 3 1 1 1 cos \cos cos 1 1 1 3 2 \dfrac{\sqrt{3}}{2} 2 3 2 2 \dfrac{\sqrt{2}}{2} 2 2 1 2 \dfrac{1}{2} 2 1 0 0 0 tan \tan tan 0 0 0 3 3 \dfrac{\sqrt{3}}{3} 3 3 1 1 1 3 \sqrt{3} 3 undefined
Memory trick for sin row: 0 , 1 2 , 2 2 , 3 2 , 1 0,\ \dfrac{1}{2},\ \dfrac{\sqrt{2}}{2},\ \dfrac{\sqrt{3}}{2},\ 1 0 , 2 1 , 2 2 , 2 3 , 1 0 , 1 , 2 , 3 , 4 \sqrt{0},\ \sqrt{1},\ \sqrt{2},\ \sqrt{3},\ \sqrt{4} 0 , 1 , 2 , 3 , 4
Fundamental trigonometric identities
Pythagorean identity:
sin 2 α + cos 2 α = 1 \sin^2\alpha + \cos^2\alpha = 1 sin 2 α + cos 2 α = 1
Derived from it:
1 + tan 2 α = 1 cos 2 α 1 + cot 2 α = 1 sin 2 α 1 + \tan^2\alpha = \frac{1}{\cos^2\alpha} \qquad 1 + \cot^2\alpha = \frac{1}{\sin^2\alpha} 1 + tan 2 α = c o s 2 α 1 1 + cot 2 α = s i n 2 α 1
Sum and difference formulas:
sin ( α ± β ) = sin α cos β ± cos α sin β \sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta sin ( α ± β ) = sin α cos β ± cos α sin β
cos ( α ± β ) = cos α cos β ∓ sin α sin β \cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta cos ( α ± β ) = cos α cos β ∓ sin α sin β
Trigonometry on the unit circle
When angle α \alpha α [ 0 ° , 90 ° ] [0°, 90°] [ 0° , 90° ]
A point on the circle has coordinates ( cos α , sin α ) (\cos\alpha,\ \sin\alpha) ( cos α , sin α )
Angles increase counterclockwise
The signs of sin/cos depend on the quadrant
Quadrant sin \sin sin cos \cos cos tan \tan tan I (0 ° 0° 0° 90 ° 90° 90° + + + + + + + + + II (90 ° 90° 90° 180 ° 180° 180° + + + − - − − - − III (180 ° 180° 180° 270 ° 270° 270° − - − − - − + + + IV (270 ° 270° 270° 360 ° 360° 360° − - − + + + − - −
Real-world applications
Indirect measurement: Know the distance and angle of elevation, calculate the height of a buildingPhysics: Decompose forces into horizontal and vertical componentsComputer graphics: Rotate objects in 2D/3D space
Example: A person stands 30 m from the base of a building. The angle of elevation to the top is 60 ° 60° 60°
h = 30 × tan 60 ° = 30 3 ≈ 51.96 m h = 30 \times \tan 60° = 30\sqrt{3} \approx 51.96 \text{ m} h = 30 × tan 60° = 30 3 ≈ 51.96 m
Practice with MathPal
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