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Determine missing angles using inverse trigonometry

Inverse trigonometric functions determine an acute angle in a right triangle from a known ratio of side lengths: sin⁡−1\sin^{-1} from opposite/hypotenuse, cos⁡−1\cos^{-1} from adjacent/hypotenuse, or tan⁡−1\tan^{-1} from opposite/adjacent. The understanding includes identifying sides relative to the unknown angle, interpreting calculator results in degrees, and recognizing that the inverse functions undo trigonometric ratios; general-angle solutions, radians, and advanced identities are not included.

Detailed Explanation: Determine missing angles using inverse trigonometry

To find a missing acute angle in a right triangle:

  1. Identify the sides relative to the unknown angle:

    • Opposite: across from the angle
    • Adjacent: next to the angle, but not the hypotenuse
    • Hypotenuse: opposite the right angle and the longest side

  2. Choose the ratio that uses the known sides:

sin⁡(θ)=oppositehypotenuse \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} cos⁡(θ)=adjacenthypotenuse \cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}} tan⁡(θ)=oppositeadjacent \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}
  1. Use the inverse function on your calculator to find the angle:
θ=sin⁡−1(ratio),θ=cos⁡−1(ratio),orθ=tan⁡−1(ratio) \theta=\sin^{-1}(\text{ratio}),\quad \theta=\cos^{-1}(\text{ratio}),\quad\text{or}\quad \theta=\tan^{-1}(\text{ratio})

Make sure your calculator is in degree mode.

Example

A right triangle has an opposite side of 77 cm and an adjacent side of 2424 cm. Find the angle θ\theta.

The known sides are the opposite and adjacent sides, so use tangent:

tan⁡(θ)=oppositeadjacent\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}

Substitute the side lengths:

tan⁡(θ)=724\tan(\theta)=\frac{7}{24}

To undo the tangent, use the inverse tangent:

θ=tan⁡−1(724)\theta=\tan^{-1}\left(\frac{7}{24}\right)

Enter this into a calculator in degree mode:

θ≈16.3∘\theta\approx 16.3^\circ

Therefore, the missing angle is:

16.3∘\boxed{16.3^\circ}

The inverse trigonometric function gives the angle that produces the known side ratio.

Learn by doing: Determine missing angles using inverse trigonometry

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