Inverse Sine and Cosine Operations


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Finding the Angle for a Sine or Cosine Ratio

Calculators have an inverse sine operation​​​​​ denoted sin-1 and an inverse cosine operation denoted cos-1. These operations will compute the angle corresponding to a given ratio to a very high precision.

Recall

Don't forget to ensure that your calculator is in degrees (DEG) mode.

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Example 4 — Angle \(a\)

In the triangles below, two of the angles \(a\), \(b\) and \(c\) are equal. Determine which angle is different from the other two.

Triangle 1 (Angle \(a\))

Right triangle with angle a. Opposite of angle a is 12 and hypotenuse is 20.

Triangle 2 (Angle \(b\))

Right triangle with angle b. Opposite of angle b is 24 and adjacent is 30.

Triangle 3 (Angle \(c\))

Right triangle with angle c. Adjacent to angle c is 28 and hypotenuse is 35.

 

 

Example 4 — Angle \(b\)

In the triangles below, two of the angles \(a\), \(b\) and \(c\) are equal. Determine which angle is different from the other two.

Triangle 1 (Angle \(a\))

Right triangle with angle a. Opposite of angle a is 12 and hypotenuse is 20.

Triangle 2 (Angle \(b\))

Right triangle with angle b. Opposite of angle b is 24 and adjacent is 30.

Triangle 3 (Angle \(c\))

Right triangle with angle c. Adjacent to angle c is 28 and hypotenuse is 35.

Solution — Angle \(b\)

Relative to the angle \(b\), we know the opposite and adjacent side lengths.

Right triangle with angle b. Opposite of angle b is 24 and adjacent is 30.

 

 

Example 4 — Angle \(c\)

In the triangles below, two of the angles \(a\), \(b\) and \(c\) are equal. Determine which angle is different from the other two.

Triangle 1 (Angle \(a\))

Right triangle with angle a. Opposite of angle a is 12 and hypotenuse is 20.

Triangle 2 (Angle \(b\))

Right triangle with angle b. Opposite of angle b is 24 and adjacent is 30.

Triangle 3 (Angle \(c\))

Right triangle with angle c. Adjacent to angle c is 28 and hypotenuse is 35.

Solution — Angle \(c\)

Relative to the angle \(c\), we know the adjacent and hypotenuse side lengths.

Right triangle with angle c. Adjacent to angle c is 28 and hypotenuse is 35.

 

Example 4 — Summary

In the triangles below, two of the angles \(a\), \(b\) and \(c\) are equal. Determine which angle is different from the other two.

Triangle 1 (Angle \(a\))

Right triangle with angle a. Opposite of angle a is 12 and hypotenuse is 20.

Triangle 2 (Angle \(b\))

Right triangle with angle b. Opposite of angle b is 24 and adjacent is 30.

Triangle 3 (Angle \(c\))

Right triangle with angle c. Adjacent to angle c is 28 and hypotenuse is 35.

Solution — Summary

To review, we have found:

\[ a \approx 36.9^{\circ} \qquad b \approx 38.7^{\circ} \qquad c \approx 36.9^{\circ}\]

 

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Example 5

Kassie needs \(70\) metres of a rope to rappel down the side of a cliff whose face is not quite vertical. The vertical distance from the base to the top of the cliff is known to be \(65\) metres.

To the nearest degree, what angle does the face of the cliff that Kassie rappels down make with the vertical?

Solution

We begin by constructing a right-angled triangle with the cliff face as the hypotenuse.

Right triangle with hypotenuse 70, angle theta, and adjacent to theta is 65 metres (height of cliff).

We label the angle between the vertical and the cliff face \(\theta\). The vertical height of the cliff is represented by the adjacent side in this triangle. This means that we know the lengths of the adjacent and hypotenuse sides. As such, we can use the inverse cosine operation to determine the angle \(\theta\).

\[ \cos (\theta) = \frac{\mathrm{adj}}{\mathrm{hyp}} \quad \implies \quad \theta = \cos^{-1} \left( \frac{\mathrm{adj}}{\mathrm{hyp}}\right)\]

Substituting in the values of the side lengths we find

\[\begin{align*} \theta &= \cos^{-1} \left( \frac{65}{70} \right) \\ \theta&\approx 21.8^{\circ} \end{align*}\]

Therefore, the angle between the cliff face and the vertical is approximately \(22^{\circ}\).


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