When to Use Sine, Cosine, or Tangent

You know the three trig ratios — sine, cosine, and tangent — and you know what SOH CAH TOA stands for. But when a problem lands in front of you, which ratio do you actually pick? All three involve the same right triangle, but each one connects a different pair of sides to the angle. Choose the wrong ratio and you waste time setting up an equation that cannot be solved, or worse, you get an answer that looks right but is not. This page gives you a systematic method so you never have to guess.

The Decision Flowchart

Every right-triangle trig problem boils down to one question: which two sides are involved? One side is known and the other is what you need to find. The flowchart below walks you through the decision.

The flowchart captures the entire decision in two questions. Start at the top, follow the branch that matches your problem, and you land on the correct ratio every time.

Quick Reference Table

Here is every scenario you will encounter when solving a right triangle with trig. The left columns tell you what the problem gives you and what it asks for; the right columns tell you which ratio to use and the rearranged formula.

Notice a pattern: sine and cosine always involve the hypotenuse. If the hypotenuse is not part of your problem, tangent is the answer.

The Three-Word Memory Aid

“SOH CAH TOA tells you which sides. Your problem tells you which ratio.”

Here is the thought process you should follow for every problem:

1. Label the sides. Relative to the angle in the problem, mark which side is Opposite (O), which is Adjacent (A), and which is the Hypotenuse (H).
2. Circle the two sides that matter. One of these is given in the problem; the other is what you need to find.
3. Match those two letters to a ratio. If your circled pair is O and H, use sine. If it is A and H, use cosine. If it is O and A, use tangent.

That is all there is to it. The mnemonic SOH CAH TOA is not just a memory trick — it is a lookup table. You are matching two letters to find the right ratio.

Worked Examples

Example 1: Ladder Problem (Sine)

A 25-foot ladder leans against a building at a 70-degree angle with the ground. How high up the building does it reach?

Step 1 — Label the sides. The ladder is the hypotenuse ($H = 25$). The height on the building is opposite the 70-degree angle. The ground distance is adjacent.

Step 2 — Identify the pair. We know H and we need O. That pair is O and H.

Step 3 — Choose the ratio. O and H means sine.

$\sin(70°) = \frac{O}{25}$

$O = 25 \times \sin(70°) \approx 25 \times 0.9397 = 23.49 \text{ ft}$

Answer: The ladder reaches approximately 23.49 feet up the building.

Example 2: Shadow and Height Problem (Tangent)

A tree casts a 30-foot shadow on flat ground. From the tip of the shadow, the angle of elevation to the top of the tree is 54 degrees. How tall is the tree?

Step 1 — Label the sides. The shadow is adjacent to the 54-degree angle ($A = 30$). The tree height is opposite. The line from the shadow tip to the treetop would be the hypotenuse, but it is not mentioned.

Step 2 — Identify the pair. We know A and we need O. That pair is O and A.

Step 3 — Choose the ratio. O and A means tangent.

$\tan(54°) = \frac{O}{30}$

$O = 30 \times \tan(54°) \approx 30 \times 1.3764 = 41.29 \text{ ft}$

Answer: The tree is approximately 41.29 feet tall.

Example 3: Ramp Problem (Inverse Trig)

A loading ramp is 20 feet long and rises 5 feet from the ground to a platform. What angle does the ramp make with the ground?

Step 1 — Label the sides. The ramp surface is the hypotenuse ($H = 20$). The rise is opposite ($O = 5$). The horizontal run is adjacent.

Step 2 — Identify the pair. We know O and H. That pair is O and H.

Step 3 — Choose the ratio. O and H means sine. Since we need the angle instead of a side, we use the inverse:

$\sin\theta = \frac{5}{20} = 0.25$

$\theta = \sin^{-1}(0.25) \approx 14.48°$

Answer: The ramp makes approximately a 14.48-degree angle with the ground.

What About Finding Angles?

When you know two sides and need the angle, the decision process is exactly the same — you still figure out which pair of sides you have (O and H, A and H, or O and A) and pick the matching ratio. The only difference is the final step: instead of solving for a side, you apply the inverse function.

• Know O and H? Use $\theta = \sin^{-1}(O/H)$
• Know A and H? Use $\theta = \cos^{-1}(A/H)$
• Know O and A? Use $\theta = \tan^{-1}(O/A)$

The choosing step does not change. Only the solving step does.

Common Mistakes

1. Forgetting to relabel sides when the angle changes. If a problem references two different angles, the opposite and adjacent sides swap when you switch angles. The hypotenuse stays the same, but always re-label O and A for the specific angle you are working with.

2. Using sine or cosine when the hypotenuse is not involved. If neither the known side nor the needed side is the hypotenuse, neither sine nor cosine will help. Use tangent.

3. Setting up the ratio upside down. Sine is $O/H$, not $H/O$. Writing the fraction backward leads to the reciprocal of the correct answer. Always put the sides in the order the mnemonic gives: SOH means O on top, H on the bottom.

4. Mixing degrees and radians. If your calculator is in radian mode and you type $\sin(70)$, you will get the sine of 70 radians — a completely different number. Make sure the mode matches the units in your problem.

Practice Problems

Test your understanding with these problems. For each one, first decide which ratio to use, then solve. Click to reveal the answer.

Key Takeaways

• Every trig problem reduces to one question: which two sides (out of O, A, H) are involved?
• If the hypotenuse is one of them: use sine (O and H) or cosine (A and H)
• If the hypotenuse is not involved: use tangent (O and A)
• Label sides first, choose ratio second — do not try to pick a ratio before you know which sides matter
• For finding angles, the choosing step is identical — only the final step changes to an inverse function
• SOH CAH TOA is your lookup table: match the pair of sides to the three-letter group that contains both letters

Return to Trigonometry for more topics in this section.