Trigonometry Reference

Right Triangles

Right-triangle trigonometry connects an acute angle to fixed side ratios. Label the triangle first, then choose the ratio that uses the sides in the problem.

Fact Table

Situation	Known Sides	Use
Find a missing side in any right triangle.	Two side lengths.	Pythagorean theorem.
Angle with opposite and hypotenuse.	$O$ and $H$	Sine.
Angle with adjacent and hypotenuse.	$A$ and $H$	Cosine.
Angle with opposite and adjacent.	$O$ and $A$	Tangent.
Find an angle from side lengths.	A trig ratio.	Inverse sine, inverse cosine, or inverse tangent.
Angles are $30^\circ$, $45^\circ$, or $60^\circ$.	One side in a special triangle.	Special right-triangle patterns.

Content Formulas

Pythagorean Theorem

$$a^2+b^2=c^2$$

SOHCAHTOA

$$\sin\theta=\frac{O}{H}\quad \cos\theta=\frac{A}{H}\quad \tan\theta=\frac{O}{A}$$

Inverse Trig

$$\theta=\sin^{-1}\left(\frac{O}{H}\right)\quad \theta=\cos^{-1}\left(\frac{A}{H}\right)\quad \theta=\tan^{-1}\left(\frac{O}{A}\right)$$

Special Right Triangles

$$45^\circ\!-\!45^\circ\!-\!90^\circ:\ x,\ x,\ x\sqrt2$$ $$30^\circ\!-\!60^\circ\!-\!90^\circ:\ x,\ x\sqrt3,\ 2x$$

The hypotenuse is always opposite the right angle. Opposite and adjacent are named relative to the acute angle being used.

Classic Examples

Label the Triangle

For the marked angle $\theta$, identify the opposite side, adjacent side, and hypotenuse.

Setup MoveFind the hypotenuse first. Then name opposite and adjacent from the marked angle.

Missing Hypotenuse

A right triangle has legs 9 and 12. Find the hypotenuse.

Solution MoveUse the Pythagorean theorem when two side lengths are known.

$$9^2+12^2=c^2$$ $$81+144=c^2$$ $$225=c^2$$ $$c=15$$

Sine for Height

A ladder is 18 feet long and makes a $62^\circ$ angle with the ground. How high does it reach?

Solution MoveUse sine because the height is opposite the angle and the ladder is the hypotenuse.

$$\sin(62^\circ)=\frac{h}{18}$$ $$h=18\sin(62^\circ)$$ $$h\approx 15.9$$

Cosine for Run

A 20-foot ramp makes a $12^\circ$ angle with the ground. Find the horizontal run.

Solution MoveUse cosine because the run is adjacent to the angle and the ramp is the hypotenuse.

$$\cos(12^\circ)=\frac{x}{20}$$ $$x=20\cos(12^\circ)$$ $$x\approx 19.6$$

Tangent for Shadow

A flagpole casts a 30-foot shadow when the angle of elevation is $38^\circ$. Find the height.

Solution MoveUse tangent because the height is opposite and the shadow is adjacent.

$$\tan(38^\circ)=\frac{h}{30}$$ $$h=30\tan(38^\circ)$$ $$h\approx 23.4$$

Inverse Angle

A right triangle has opposite side 7 and hypotenuse 12 relative to $\theta$. Find $\theta$.

Solution MoveUse inverse trig when the angle is unknown and the side ratio is known.

$$\sin\theta=\frac{7}{12}$$ $$\theta=\sin^{-1}\left(\frac{7}{12}\right)$$ $$\theta\approx 35.7^\circ$$

45-45-90 Diagonal

A square has side length 9. Find the diagonal.

Solution MoveA square diagonal creates a 45-45-90 triangle, so multiply a leg by $\sqrt2$.

$$d=9\sqrt2$$

30-60-90 Triangle

A 30-60-90 triangle has hypotenuse 16. Find the short leg and long leg.

Solution MoveThe hypotenuse is twice the short leg. The long leg is the short leg times $\sqrt3$.

$$2x=16$$ $$x=8$$ $$x\sqrt3=8\sqrt3$$