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Trig Functions

Trig Functions
Circular Functions

The six functions sine, cosine, tangent, cosecant, secant, and cotangent. These functions can be defined several different ways. These include SOHCAHTOA definitions, circle definitions (below), and unit circle definitions.

 

Circle with point (x,y) on radius r; defines sin=y/r, cos=x/r, tan=y/x, csc=r/y, sec=r/x, cot=x/y.

 

 

See also

Inverse trig functions

Key Formula

sin⁡θ=oppositehypotenuse,cos⁡θ=adjacenthypotenuse,tan⁡θ=oppositeadjacent\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}} csc⁡θ=1sin⁡θ,sec⁡θ=1cos⁡θ,cot⁡θ=1tan⁡θ\csc\theta = \frac{1}{\sin\theta}, \quad \sec\theta = \frac{1}{\cos\theta}, \quad \cot\theta = \frac{1}{\tan\theta}
Where:
  • θ\theta = The angle of interest in the right triangle or on the unit circle
  • opposite\text{opposite} = The side across from angle θ in a right triangle
  • adjacent\text{adjacent} = The side next to angle θ (not the hypotenuse) in a right triangle
  • hypotenuse\text{hypotenuse} = The longest side of the right triangle, opposite the right angle

Worked Example

Problem: A right triangle has an angle θ, an opposite side of length 3, an adjacent side of length 4, and a hypotenuse of length 5. Find all six trig functions of θ.
Step 1: Identify the three sides relative to angle θ: opposite = 3, adjacent = 4, hypotenuse = 5.
Step 2: Compute the three primary trig functions using SOHCAHTOA.
sin⁡θ=35,cos⁡θ=45,tan⁡θ=34\sin\theta = \frac{3}{5}, \quad \cos\theta = \frac{4}{5}, \quad \tan\theta = \frac{3}{4}
Step 3: Find the three reciprocal trig functions by flipping each fraction.
csc⁡θ=53,sec⁡θ=54,cot⁡θ=43\csc\theta = \frac{5}{3}, \quad \sec\theta = \frac{5}{4}, \quad \cot\theta = \frac{4}{3}
Step 4: Verify: check that tan θ = sin θ / cos θ.
sin⁡θcos⁡θ=3/54/5=34=tan⁡θ  ✓\frac{\sin\theta}{\cos\theta} = \frac{3/5}{4/5} = \frac{3}{4} = \tan\theta \; \checkmark
Answer: sin θ = 3/5, cos θ = 4/5, tan θ = 3/4, csc θ = 5/3, sec θ = 5/4, cot θ = 4/3.

Another Example

This example uses the unit circle definition instead of a right triangle, and involves an angle in Quadrant II where some trig values are negative.

Problem: Find the exact values of the six trig functions for θ = 5π/6 using the unit circle.
Step 1: Recognize that 5π/6 is in Quadrant II. Its reference angle is π − 5π/6 = π/6.
Reference angle=π6\text{Reference angle} = \frac{\pi}{6}
Step 2: Recall the unit circle coordinates for π/6: (√3/2, 1/2). In Quadrant II, x is negative and y is positive.
(−32,  12)\left(-\frac{\sqrt{3}}{2},\; \frac{1}{2}\right)
Step 3: On the unit circle, cos θ = x-coordinate and sin θ = y-coordinate.
cos⁡5π6=−32,sin⁡5π6=12\cos\frac{5\pi}{6} = -\frac{\sqrt{3}}{2}, \quad \sin\frac{5\pi}{6} = \frac{1}{2}
Step 4: Compute the remaining four functions from sine and cosine.
tan⁡5π6=1/2−3/2=−13=−33\tan\frac{5\pi}{6} = \frac{1/2}{-\sqrt{3}/2} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}
Step 5: Take reciprocals to get cosecant, secant, and cotangent.
csc⁡5π6=2,sec⁡5π6=−233,cot⁡5π6=−3\csc\frac{5\pi}{6} = 2, \quad \sec\frac{5\pi}{6} = -\frac{2\sqrt{3}}{3}, \quad \cot\frac{5\pi}{6} = -\sqrt{3}
Answer: sin(5π/6) = 1/2, cos(5π/6) = −√3/2, tan(5π/6) = −√3/3, csc(5π/6) = 2, sec(5π/6) = −2√3/3, cot(5π/6) = −√3.

Frequently Asked Questions

What is the easiest way to remember the six trig functions?
Use the mnemonic SOHCAHTOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. For the other three, remember that cosecant is the reciprocal of sine, secant is the reciprocal of cosine, and cotangent is the reciprocal of tangent. A helpful memory trick: each 'co-' function is the reciprocal of the function without 'co-', except for cosine and secant, which you just need to memorize.
What is the difference between trig functions and inverse trig functions?
Trig functions take an angle as input and return a ratio (e.g., sin 30° = 0.5). Inverse trig functions do the reverse: they take a ratio as input and return an angle (e.g., sin⁻¹(0.5) = 30°). Inverse trig functions are written as sin⁻¹, cos⁻¹, tan⁻¹ or equivalently arcsin, arccos, arctan.
When should you use radians versus degrees for trig functions?
Both radians and degrees work for evaluating trig functions, and the outputs are the same — sin(π/6) and sin(30°) both equal 1/2. However, in calculus and most higher math, radians are required because derivative and integral formulas for trig functions assume radian measure. Always check which mode your calculator is set to before computing.

Trig Functions vs. Inverse Trig Functions

Trig FunctionsInverse Trig Functions
InputAn angle (e.g., 45° or π/4)A ratio or number (e.g., 0.5)
OutputA ratio or numberAn angle
Notationsin θ, cos θ, tan θsin⁻¹x, cos⁻¹x, tan⁻¹x (or arcsin, arccos, arctan)
Examplesin(30°) = 0.5sin⁻¹(0.5) = 30°
When to useFinding a side length or ratio given an angleFinding an unknown angle given side lengths

Why It Matters

Trig functions appear throughout high school math — from solving right triangles in geometry to graphing sinusoidal waves in precalculus to computing derivatives and integrals in calculus. They are equally important in physics for modeling waves, circular motion, and forces on inclined planes. Mastering all six functions and their relationships is essential for success on standardized tests like the SAT and ACT.

Common Mistakes

Mistake: Confusing which sides are 'opposite' and 'adjacent' — these labels change depending on which angle you are looking at.
Correction: Always identify opposite and adjacent relative to the specific angle θ in the problem. The hypotenuse is always across from the 90° angle, but opposite and adjacent swap if you switch to the other acute angle.
Mistake: Mixing up the reciprocal pairs: thinking secant is the reciprocal of sine, or cosecant is the reciprocal of cosine.
Correction: Cosecant (csc) is the reciprocal of sine, and secant (sec) is the reciprocal of cosine. The pairing is counterintuitive because the 'co-' names don't match up. Memorize: csc goes with sin, sec goes with cos, cot goes with tan.

Related Terms