Identity

Pythagorean trigonometric identities

Trigonometry • Trigonometric Identity

Formula

sin^{2} θ + cos^{2} θ = 1

For all

θ

where each side is defined:

sin^{2} θ + cos^{2} θ = 1

1 + tan^{2} θ = sec^{2} θ

, and

1 + cot^{2} θ = csc^{2} θ

How to use it

1Use to eliminate one ratio in favour of another.
2The second and third follow from the first on dividing by $cos^{2} θ$ and $sin^{2} θ$ .
3 $sec^{2} θ - tan^{2} θ = 1$ factorises as $(sec θ + tan θ) (sec θ - tan θ) = 1$ , which answers many reciprocal-pair questions.

sin θ = \frac{3}{5}

and

θ

is acute, find

cos θ

Step-by-step

1
$cos^{2} θ = 1 - sin^{2} θ = 1 - \frac{9}{25} = \frac{16}{25}$
Why: Apply the identity.
2
$cos θ = \frac{4}{5}$
Why: Acute, so the positive root is taken.

Answer

\frac{4}{5}

Watch out

Taking the square root reintroduces a sign; the quadrant decides it.

tan

and

sec

forms are undefined at

θ = 9 0^{\circ}

Builds on

Also known as

sin^2 + cos^2 = 1, fundamental trig identity