Finding Missing Angles

In the previous topic, you used a known angle and one side to find a missing side. Now we reverse the process: given two sides, find the missing angle. This requires inverse trigonometric functions — sometimes called “arc” functions.

What Are Inverse Trig Functions?

A regular trig function takes an angle and returns a ratio. An inverse trig function takes a ratio and returns an angle.

The notation $\sin^{-1}$ does not mean $\frac{1}{\sin}$. It means “the angle whose sine is.” On most calculators, you access these functions by pressing the 2nd or SHIFT key before pressing SIN, COS, or TAN.

The Process

Step 1 — Label the sides (O, A, H) relative to the unknown angle.

Step 2 — Form the ratio. Divide the two known sides to create the appropriate trig ratio.

Step 3 — Apply the inverse function. Use your calculator to find the angle.

Worked Examples

Example 1: Finding an Angle Using Inverse Sine

A right triangle has an opposite side of 6 and a hypotenuse of 10. Find the angle $\theta$.

We know O and H, so we use sine:

$\sin\theta = \frac{O}{H} = \frac{6}{10} = 0.6$

$\theta = \sin^{-1}(0.6) = 36.87°$

Answer: $\theta \approx 36.87°$

Example 2: Finding an Angle Using Inverse Tangent

A right triangle has an opposite side of 5 and an adjacent side of 9. Find the angle $\theta$.

We know O and A, so we use tangent:

$\tan\theta = \frac{O}{A} = \frac{5}{9}$

$\theta = \tan^{-1}\!\left(\frac{5}{9}\right) = 29.05°$

Answer: $\theta \approx 29.05°$

Example 3: Finding Both Acute Angles

A right triangle has legs of 8 and 15 and a hypotenuse of 17. Find both acute angles.

Angle A (opposite the side of length 8):

$\sin A = \frac{8}{17} = 0.4706$

$A = \sin^{-1}(0.4706) = 28.07°$

Angle B (opposite the side of length 15):

$B = 90° - 28.07° = 61.93°$

Verification: $\sin(61.93°) = \frac{15}{17} = 0.8824$, and $\sin^{-1}(0.8824) = 61.93°$. Correct.

Answer: The two acute angles are approximately 28.07° and 61.93°.

Notice that the two acute angles always add to 90 degrees. This gives you a shortcut: find one angle, then subtract from 90 to get the other.

Calculator Tips

1. Check your mode. Your calculator must be in degree mode (not radian mode) if you want the answer in degrees. Look for “DEG” on the display.
2. Access inverse functions. On most scientific calculators, press 2nd (or SHIFT or INV), then the trig button.
3. Valid input ranges:
   • $\sin^{-1}$ and $\cos^{-1}$ accept values from $-1$ to $1$ only
   • $\tan^{-1}$ accepts any real number
4. If you get an error, check that your ratio is within the valid range. A sine or cosine value greater than 1 or less than $-1$ is impossible — recheck your side labels.

Real-World Application: Electrician — Conduit Bend Angle

An electrician needs to run conduit from a horizontal ceiling to a panel that is 4 feet lower and offset 7 feet horizontally. They need to calculate the bend angle for the conduit.

The vertical drop (4 ft) is opposite the bend angle. The horizontal offset (7 ft) is adjacent.

Step 1 — Choose the ratio. We know O and A. Use tangent.

$\tan\theta = \frac{4}{7} = 0.5714$

Step 2 — Find the angle.

$\theta = \tan^{-1}(0.5714) = 29.74°$

Step 3 — Apply to the conduit bend. The electrician sets their conduit bender to approximately 30 degrees. They can also calculate the conduit length (the hypotenuse):

$H = \frac{4}{\sin(29.74°)} = \frac{4}{0.4961} = 8.06 \text{ ft}$

Answer: The bend angle is approximately 30 degrees and the conduit run is about 8.06 feet.

Finding the Third Angle

In any triangle, the three angles add up to 180 degrees. In a right triangle, one angle is always 90 degrees, so:

$\text{Angle A} + \text{Angle B} = 90°$

This means you only ever need to calculate one acute angle — the other is $90° - \theta$.

Common Mistakes

1. Calculator in radian mode. If you get strange decimal answers like 0.6435 instead of 36.87, your calculator is in radian mode. Switch to degrees.
2. Confusing $\sin^{-1}$ with $\frac{1}{\sin}$. These are completely different operations. $\sin^{-1}(0.5) = 30°$ but $\frac{1}{\sin(0.5°)} = 114.59$.
3. Using the wrong sides. Double-check that you have identified O, A, and H correctly relative to the angle you want to find.
4. Forgetting the right angle. You only need to calculate one acute angle; the other is $90° - \theta$.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Inverse trig functions take a ratio and return an angle: $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$
• The notation $\sin^{-1}(x)$ means “the angle whose sine is $x$” — not $1/\sin(x)$
• Always check that your calculator is in degree mode when working with degrees
• In a right triangle, the two acute angles always sum to 90 degrees — find one and subtract to get the other
• Inverse trig is essential in the trades: electricians use it for bend angles, carpenters for pitch angles, and surveyors for measuring angles of elevation

For a deeper treatment of inverse trig functions as mathematical functions — restricted domains, graphs, and evaluating compositions like $\sin(\arccos x)$ — see Inverse Trigonometric Functions.

Return to Trigonometry for more topics in this section.