What does SOHCAHTOA stand for?

Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. It is a memory hook for choosing the ratio that connects the side you know to the side you want.

How do I know which side is opposite and which is adjacent?

Always relative to the angle you are using. The hypotenuse is fixed opposite the right angle. Of the two legs, the one touching your angle is adjacent and the one across from it is opposite — and they swap if you switch to the other acute angle.

When do I use the inverse functions?

When the angle is what you are looking for. If two sides are known, form the ratio and apply \sin^{-1}, \cos^{-1} or \tan^{-1} to it. Going the other way — angle known, side wanted — uses the plain functions.

Does this work on triangles without a right angle?

No. SOHCAHTOA depends on the right angle to define the hypotenuse. For other triangles you need the law of sines or the law of cosines, which work for any triangle at all.

What is the difference between angle of elevation and angle of depression?

Elevation is measured upward from the horizontal, depression downward from it. Between the same two points they are equal (alternate angles), which is why a depression problem can be redrawn as an elevation problem at the other end.

Can Photo Math Solver read the triangle from a diagram?

Yes — screenshot the figure with its labels, angle marks, and the right-angle square. The solver works out which side is opposite the marked angle before it chooses a ratio, which is the step most answers turn on.

You can start solving right triangles for free by adding the Chrome extension — try it on your own worksheet before deciding if you need anything more.

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Photo Math Solver Solved

\sin 30° = \frac{x}{10}

1Multiply both sides by $10$ : $x = 10\sin 30°$ .
2 $\sin 30° = 0.5$ , so $x = 5$ .

On this page01Definition 02Formulas 03Methods 04Examples 05Practice 06Mistakes 07FAQ

01Definition

What is right triangle trigonometry?

Right-triangle trigonometry uses three ratios to connect an acute angle to the sides of a right triangle. Sine, cosine and tangent each compare a specific pair of sides, and because all right triangles sharing an angle are similar, each ratio depends only on the angle — never on the triangle’s size.

Which side is which depends on the angle you are working from. The hypotenuse is fixed — always opposite the right angle — but "opposite" and "adjacent" swap over the moment you switch to the other acute angle. Labelling the sides relative to the marked angle, before choosing a ratio, is most of the work.

SOHCAHTOA is the memory hook: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. Run it forwards to find a missing side from a known angle; run it backwards with $\sin^{-1}$ , $\cos^{-1}$ or $\tan^{-1}$ to find a missing angle from two known sides.

Where you'll actually use this

Surveying building heights from a ground distance and an angle of elevation.
Ramp and roof pitch: an accessibility ramp’s angle comes straight from rise over length.
Navigation and bearings, where a heading and distance resolve into north and east components.
Physics force diagrams: splitting a force on a slope into components parallel and perpendicular to it.
Camera framing and line of sight — working out how far back to stand to fit a subject in view.

Hypotenuse: The side opposite the right angle — always the longest.
Opposite side: The leg across from the angle you are working with.
Adjacent side: The leg that touches the angle you are working with (and is not the hypotenuse).
Inverse trig function: $\sin^{-1}$ , $\cos^{-1}$ , $\tan^{-1}$ — return the angle that produces a given ratio.
Angle of elevation: The angle measured upward from the horizontal to a line of sight.

02Formulas

The formulas you need

Keep these on hand — every method below builds on them.

Sine

\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}

The SOH of SOHCAHTOA.

Cosine

\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}

The CAH — adjacent is the leg that touches the angle.

Tangent

\tan\theta = \frac{\text{opposite}}{\text{adjacent}}

The TOA — the only ratio with no hypotenuse in it.

Finding an angle

\theta = \sin^{-1}\left(\frac{o}{h}\right)

Inverse functions; $\sin^{-1}$ means "the angle whose sine is", not $1/\sin$.

Special angles

\sin 30° = \tfrac{1}{2},\; \cos 60° = \tfrac{1}{2},\; \tan 45° = 1

Exact values worth memorising — they turn up constantly.

Angle sum

\theta_1 + \theta_2 = 90°

The two acute angles are complementary, so one known angle gives the other for free.

03Methods

How to solve right triangle trigonometry

Pick the method that fits the problem in front of you.

When to use it: One acute angle and one side are given, and another side is the unknown.

1Label the sides relative to the marked angle: hypotenuse, opposite, adjacent.
2Pick the ratio that uses the side you have and the side you want — for opposite and hypotenuse, that is sine.
3Write the equation and substitute: $\sin 30° = \frac{x}{10}$ .
4Rearrange: $x = 10\sin 30° = 5$ . If the unknown is in the denominator, multiply up then divide.

04Worked examples

Right Triangle Trigonometry, solved step by step

From easy to hard — pick a problem to see its full solution.

\sin 30° = \frac{x}{10}

1Multiply both sides by $10$ : $x = 10\sin 30°$ .
2 $\sin 30° = 0.5$ , so $x = 5$ .

Try it on your own right triangle trigonometry homework

Photo Math Solver reads a right-triangle problem straight from a screenshot — a labelled diagram, an angle-of-elevation word problem, a worksheet asking for all missing parts — and shows the full working: which side is opposite and which is adjacent to the marked angle, which of sine, cosine or tangent that makes correct, and the rearrangement to the answer.

05Practice

Now you try

Work each one on paper first, then check your answer.

Problem 1

\tan 45° = \frac{x}{8}

Problem 2

\theta = ?, \quad \text{adjacent} = 9,\; \text{hypotenuse} = 15

Problem 3

\sin 25° = \frac{6}{h}

Problem 4

\text{Find the opposite side: } \theta = 58°,\; \text{adjacent} = 11

Problem 5

\text{A 3 m ramp rises 0.5 m. Find its angle of elevation.}

Stuck on your right triangle trigonometry homework? Screenshot it

Right-triangle questions are decided by which side is opposite the marked angle — information that lives on the diagram, not in the sentence. Screenshot the whole figure with Photo Math Solver and the solution labels the sides first, then explains why it picked sine, cosine, or tangent.

06Common mistakes

Where points get lost

Each of these shows up on real graded work — and each has a simple fix.

The mistake	Why it happens	The fix
Leaving the calculator in radian mode and getting answers like $\sin 30° = -0.988$ .	The mode persists from a previous problem, and the wrong answer still looks like a plausible number.	Check for DEG or RAD on the display before starting. A quick test: $\sin 30°$ must give exactly $0.5$ in degree mode.
Mixing up opposite and adjacent after switching to the other acute angle.	The sides get labelled once for the first angle and are not relabelled when the question moves on.	Re-label from scratch for each angle you work with. Only the hypotenuse stays put; opposite and adjacent trade places.
Reading $\sin^{-1}x$ as $\frac{1}{\sin x}$ .	The $-1$ superscript means reciprocal everywhere else in algebra, so the notation genuinely misleads.	$\sin^{-1}$ is the inverse function — "the angle whose sine is". The reciprocal of sine is cosecant, written $\csc$ , which is a different thing entirely.
Dividing when the unknown is in the denominator: getting $h = 12\cos 40°$ from $\cos 40° = \frac{12}{h}$ .	The rearrangement pattern from the numerator case gets reused without checking where the unknown actually sits.	Multiply both sides by $h$ first, then divide by the ratio: $h = \frac{12}{\cos 40°}$ . The answer must be larger than $12$ , since the hypotenuse is the longest side.
Using trigonometry when two sides are known and no angle is involved.	The chapter is about trig, so every problem in it looks like a trig problem.	Two sides in, third side out, no angles anywhere — that is the Pythagorean theorem. Reach for a trig ratio only when an angle is either given or wanted.

Photo Math Solver shows every intermediate step, so slips like these are easy to catch before they cost you marks.

07FAQ