Right Triangle Trigonometry

What “Solving” Means

To solve a triangle is to find every missing side and angle from the few pieces you already know.

A right triangle makes this easy. One angle is already $90°$, so if you know just two more facts (a side and an angle, or two sides), trigonometry hands you the rest.

The Three Ratios

Everything rests on how the three sides relate to an angle $\theta$:

• hypotenuse: the long side, always opposite the right angle
• opposite: the side across from $\theta$
• adjacent: the side next to $\theta$ (that is not the hypotenuse)

As the angle opens up, the opposite side grows and the adjacent side shrinks, and the three ratios track it exactly. Those ratios are sine, cosine, and tangent.

$\sin\theta = \frac{\text{opp}}{\text{hyp}} \qquad \cos\theta = \frac{\text{adj}}{\text{hyp}} \qquad \tan\theta = \frac{\text{opp}}{\text{adj}}$

SOH-CAH-TOA

The classic way to remember them:

Pick whichever ratio connects the side you know to the side you want.

Finding a Side

A ramp rises at $40°$, and its slope (the hypotenuse) is $10$ m long. How high does it reach?

The height is the opposite side, and we know the hypotenuse. Opposite and hypotenuse means sine:

Finding an Angle

When you know two sides but want the angle, run the ratio backwards with an inverse trig function.

A right triangle has opposite $3$ and adjacent $4$. What is the angle?

Opposite and adjacent means tangent:

Know two sides, want the angle? Use the matching inverse function.

Choosing the Right Ratio

Match what you have to what you want:

A Full Solve

Given a right triangle with angle $30°$ and adjacent side $8$. Find everything else.

The other angle (angles sum to $180°$, one is $90°$):

$180° - 90° - 30° = 60°$

The opposite side (angle + adjacent, want opposite, so tangent):

$\text{opp} = 8\tan 30° \approx 4.62$

The hypotenuse (angle + adjacent, want hypotenuse, so cosine):

$\text{hyp} = \frac{8}{\cos 30°} \approx 9.24$

Three known facts became a fully solved triangle.