Finding Missing Angles with Inverse Trig: sin⁻¹, cos⁻¹, tan⁻¹

Inverse Trig: the “Undo” Button for Ratios

In a right triangle, sin, cos, and tan take an angle and return a ratio of side lengths. Inverse trig does the reverse: it takes a ratio and returns the angle that produced it.

When you know two sides (so you can form a ratio), you can recover the acute angle θ using:

• θ = sin⁻¹(opposite / hypotenuse)
• θ = cos⁻¹(adjacent / hypotenuse)
• θ = tan⁻¹(opposite / adjacent)

Important notation warning: sin⁻¹ is NOT 1/sin

sin⁻¹(x) means “the angle whose sine is x.” It does not mean 1/sin(x). Compare:

• sin⁻¹(0.5) = 30° (an angle)
• 1/sin(30°) = 2 (a number, the reciprocal)

If you ever see csc(θ), that is the reciprocal of sine: csc(θ)=1/sin(θ). That is different from sin⁻¹.

Calculator Setup and Input (Parentheses Matter)

1) Degree mode

For right-triangle problems in this course, angles are in degrees. Make sure your calculator is in DEG mode (not RAD).

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2) Use parentheses around the ratio

You want the inverse trig function to act on the entire fraction.

Correct:  sin⁻¹( 7/10 )   or   sin⁻¹(0.7)  → angle in degrees

Incorrect: sin⁻¹ 7 / 10   (this can be interpreted as (sin⁻¹(7))/10)

3) Typical button sequences

Many calculators use an SHIFT or 2nd key for inverse trig:

• SHIFT + sin gives sin⁻¹(
• SHIFT + cos gives cos⁻¹(
• SHIFT + tan gives tan⁻¹(

Example 1: Find an Angle Using sin⁻¹ (Opposite and Hypotenuse Known)

Suppose a right triangle has an acute angle θ with:

• opposite side = 7
• hypotenuse = 10

Step 1: Build the ratio.

sin(θ) = opposite/hypotenuse = 7/10 = 0.7

Step 2: “Undo” sine with inverse sine.

θ = sin⁻¹(0.7)

Step 3: Calculator input.

θ ≈ sin⁻¹(0.7) ≈ 44.4° (your calculator may show 44.427…)

Quick check: since 0.7 is between 0 and 1, it is a valid sine ratio for an acute angle, and 44.4° is between 0° and 90°, so it makes sense.

Find the other acute angle

In a right triangle, the two acute angles add to 90°. If the other acute angle is φ, then:

φ = 90° − θ = 90° − 44.4° = 45.6°

Reasonableness check: if one acute angle is about 44°, the other being about 46° is plausible (they should be complementary).

Example 2: Find an Angle Using cos⁻¹ (Adjacent and Hypotenuse Known)

Suppose for an acute angle θ you know:

• adjacent side = 12
• hypotenuse = 13

Step 1: Build the ratio.

cos(θ) = adjacent/hypotenuse = 12/13 ≈ 0.9230769

Step 2: Apply inverse cosine.

θ = cos⁻¹(12/13)

Step 3: Evaluate.

θ ≈ cos⁻¹(0.9230769) ≈ 22.6°

Find the remaining acute angle

φ = 90° − 22.6° = 67.4°

Quick check: 12/13 is between 0 and 1, so it is a valid cosine ratio for an acute angle, and the result is between 0° and 90°.

Example 3: Find an Angle Using tan⁻¹ (Opposite and Adjacent Known)

Suppose for an acute angle θ you know:

• opposite side = 9
• adjacent side = 4

Step 1: Build the ratio.

tan(θ) = opposite/adjacent = 9/4 = 2.25

Step 2: Apply inverse tangent.

θ = tan⁻¹(9/4)

Step 3: Evaluate.

θ ≈ tan⁻¹(2.25) ≈ 66.0°

Find the remaining acute angle

φ = 90° − 66.0° = 24.0°

Quick check: tangent ratios for acute angles are positive, and can be greater than 1, so 2.25 is fine. The angle is still between 0° and 90°, so it fits a right triangle’s acute angle.

Error-Checking Routines (Catch Mistakes Fast)

1) Angle range check for right-triangle acute angles

If you are solving for an acute angle in a right triangle, your answer should satisfy:

0° < θ < 90°

If you get 0°, 90°, a negative angle, or something above 90°, check for one of these issues:

• calculator in RAD mode instead of DEG mode
• wrong ratio (mixed up which sides you used)
• missing parentheses in the calculator input

2) Valid bounds check for sine and cosine ratios

For any real triangle side lengths, the ratios for sine and cosine must be between 0 and 1 for acute angles:

• 0 < opposite/hypotenuse < 1 (sine input)
• 0 < adjacent/hypotenuse < 1 (cosine input)

If your computed ratio is greater than 1 (or negative when all side lengths are positive), something is wrong—often a swapped numerator/denominator or using the hypotenuse in the wrong place.

3) Complement check (two acute angles sum to 90°)

After finding one acute angle θ, compute the other as 90° − θ. Then do a quick sanity check:

• Both angles should be between 0° and 90°.
• The larger angle should correspond to the larger opposite side (qualitative check).

4) Re-check by forward trig (optional but powerful)

Take your computed angle and plug it back into the original trig function to see if you recover the ratio (approximately, due to rounding).

Example from above: if θ ≈ 44.4°, then sin(44.4°) ≈ 0.7. If it’s far off, re-check mode and input.