The trigonometric ratios sine, cosine, and tangent each relate an angle to a pair of sides in a right triangle. If we know an acute angle measure in the right triangle and a side length, we can solve for another side length of the triangle using sine, cosine, or tangent as follows:

- Highlight the reference angle.
- Identify which sides are the hypotenuse, opposite, and adjacent - label if desired.
- Determine which trigonometric ratio to use. Choose the ratio for which we have one of the required sides given and one is the missing side:
- Sine: opposite and hypotenuse
- Cosine: adjacent and hypotenuse
- Tangent: opposite and adjacent

- Set up the chosen ratio: \sin\theta=\dfrac{\text{opposite}}{\text{hypotenuse}} \qquad \cos\theta=\dfrac{\text{adjacent}}{\text{hypotenuse}} \qquad \tan\theta=\dfrac{\text{opposite}}{\text{adjacent}}
- Solve for the variable using inverse operations.

Remember the following:

- For a given angle \theta, the value of each trigonometric ratio stays the same no matter the size of the triangle
- Your calculator must be set to degree mode when working with degrees in trigonometric ratios

Find the missing side length for each triangle. Round your answer to two decimal places.

a

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b

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A girl is flying a kite that is attached to the end of a 23.4 \text{ m} length of string. The angle between the string and the vertical is 21 \degree. If the girl is holding the string 2.1 \text{ m} above the ground, find the vertical distance from the ground to the place where the string attaches to the kite. Round your answer to two decimal places.

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Find the value of x, the side length of the parallelogram, rounding your answer to the nearest inch.

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Idea summary

If we know an acute angle measure in the right triangle and a side length, we can solve for another side length of the triangle using sine, cosine, or tangent:

- Highlight the reference angle.
- Identify which sides are the hypotenuse, opposite, and adjacent - label if desired.
- Determine which trigonometric ratio to use. Choose the ratio for which we have one of the required sides given and one is the missing side:
- Sine: opposite and hypotenuse
- Cosine: adjacent and hypotenuse
- Tangent: opposite and adjacent

- Set up the chosen ratio: \sin\theta=\dfrac{\text{opposite}}{\text{hypotenuse}} \qquad \cos\theta=\dfrac{\text{adjacent}}{\text{hypotenuse}} \qquad \tan\theta=\dfrac{\text{opposite}}{\text{adjacent}}
- Solve for the variable using inverse operations.

The trigonometric ratios take angles and give side ratios, but we are going to need functions that take side ratios and give angles.

We are already familiar with certain operations having inverse operations that "undo" them - such as addition and subtraction, or multiplication and division. In a similar manner, we can apply **inverse trigonometric functions** to reverse, or "undo", the existing trigonometric functions.

The functions sine, cosine, and tangent each take an angle measure as input and return a side ratio as output.

For a given angle \theta we have:\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \qquad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \qquad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}

The inverse functions work in reverse, taking in the ratio of two known sides as the input and returning a missing angle measure as the output.

So the three inverse trigonometric functions are: \sin^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)= \theta \qquad \cos^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right) = \theta \qquad \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right) = \theta

We can use these inverse trigonometric functions to determine an unknown angle measure when we know a pair of side lengths of a right-triangle.

As these inverse functions "undo" the original functions, we have\sin^{-1}\left(\sin\left(x\right)\right) = x \qquad \cos^{-1}\left(\cos\left(x\right)\right) = x \qquad \tan^{-1}\left(\tan\left(x\right)\right) = xand\sin\left(\sin^{-1}\left(x\right)\right) = x \qquad \cos\left(\cos^{-1}\left(x\right)\right) = x \qquad \tan\left(\tan^{-1}\left(x\right)\right) = x

It is important to note that while the notation used to represent these inverse trigonometric functions looks like a power of -1, these functions are *not* the same as the reciprocals of the trigonometric functions. That is:\sin^{-1}(x) \neq \frac{1}{\sin(x)} \qquad \cos^{-1}(x) \neq \frac{1}{\cos(x)} \qquad \tan^{-1}(x) \neq \frac{1}{\tan(x)}

Explore the applet by dragging the hypotenuse of the right triangle.

- Change the hypotenuse to match the desired ratio. Calculate the inverse trigonometric ratio, then check the box to show the angle. Does your calculated measurement match the actual angle measure?
- Repeat by clicking New Goal.

If \cos \theta=0.256, find \theta. Round your answer to two decimal places.

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Solve for the value of x. Round your answer to two decimal places.

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A ladder of length 3.46 \text{ m} needs to make an angle with the wall of between 10 \degree and 20 \degree for it to be safe to stand on. At steeper angles, the ladder is at risk of toppling backwards when the climber leans away from it and at shallower angles, the ladder may lose its grip on the ground.

The base of the ladder is placed at 190 \text{ cm}, 42 \text{ cm}, or 86 \text{ cm} from the wall. At which of these base lengths is the ladder safest to climb? Explain how you found your answer.

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An isosceles triangle has equal side lengths of 10 \text{ in} and a base of 8 \text{ in} as shown.

Find m\angle A, to the nearest tenth of a degree.

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Idea summary

We can use these **inverse trigonometric functions** to determine an unknown angle measure when we know a pair of side lengths of a right-triangle: \sin^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)= \theta \qquad \cos^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right) = \theta \qquad \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right) = \theta