المتطابقات التبادلية

csc

$\csc\theta = \frac{1}{\sin\theta}$

قاطع التمام هو مقلوب الجيب.

sec

$\sec\theta = \frac{1}{\cos\theta}$

القاطع هو مقلوب جيب التمام.

cot

$\cot\theta = \frac{1}{\tan\theta}$

ظل التمام هو مقلوب الظل.

متطابقات القسمة

tan من sin/cos

$\tan\theta = \frac{\sin\theta}{\cos\theta}$

حوّل tan دائمًا إلى sin/cos عند التعثر.

cot من cos/sin

$\cot\theta = \frac{\cos\theta}{\sin\theta}$

ظل التمام معبَّرًا عنه بالدالتين الأساسيتين.

Pythagorean identities

Main identity

$\sin^2\theta + \cos^2\theta = 1$

The most-used identity in trigonometry — derives from the unit circle.

Tan form

$1 + \tan^2\theta = \sec^2\theta$

Divide the main identity by $\cos^2\theta$ .

Cot form

$1 + \cot^2\theta = \csc^2\theta$

Divide the main identity by $\sin^2\theta$ .

Even-odd / cofunction

sin is odd

$\sin(-\theta) = -\sin\theta$

Sin reflects across the origin.

cos is even

$\cos(-\theta) = \cos\theta$

Cos reflects across the y-axis.

tan is odd

$\tan(-\theta) = -\tan\theta$

Inherits oddness from sin/cos.

Cofunction (sin)

$\sin\bigl(\tfrac{\pi}{2} - \theta\bigr) = \cos\theta$

Sine of complementary angle = cosine of original.

Cofunction (tan)

$\tan\bigl(\tfrac{\pi}{2} - \theta\bigr) = \cot\theta$

Tan-cot pair.

Sum and difference

sin sum

$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$

The single most useful sum formula.

cos sum

$\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$

Note the flipped sign in the result.

tan sum

$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

Useful when working purely in tangents.

Double-angle

sin 2θ

$\sin(2\theta) = 2\sin\theta\cos\theta$

Comes from sum identity with $A = B = \theta$ .

cos 2θ (three forms)

$\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$

Pick the form that matches what you have.

tan 2θ

$\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}$

Avoid when $\tan\theta = \pm 1$ (undefined).

Half-angle

sin half-angle

$\sin\bigl(\tfrac{\theta}{2}\bigr) = \pm\sqrt{\tfrac{1 - \cos\theta}{2}}$

Sign depends on quadrant of $\theta/2$ .

cos half-angle

$\cos\bigl(\tfrac{\theta}{2}\bigr) = \pm\sqrt{\tfrac{1 + \cos\theta}{2}}$

Same caveat about sign.

tan half-angle

$\tan\bigl(\tfrac{\theta}{2}\bigr) = \frac{1 - \cos\theta}{\sin\theta} = \frac{\sin\theta}{1 + \cos\theta}$

Two equivalent forms — pick whichever avoids division by zero.

Product-to-sum (advanced)

sin·cos

$\sin A \cos B = \tfrac{1}{2}[\sin(A+B) + \sin(A-B)]$

Converts product to sum — useful in integrals.

cos·cos

$\cos A \cos B = \tfrac{1}{2}[\cos(A-B) + \cos(A+B)]$

Same role for cosines.

sin·sin

$\sin A \sin B = \tfrac{1}{2}[\cos(A-B) - \cos(A+B)]$

Note the negative sign.

علم المثلثات Formulas