A **trigonometric ratio**, or **trigonometric function**, is a relationship between an angle and a pair of sides in a right triangle.

To talk about the trigonometric ratios, we first label the sides of a right triangle with respect to a particular angle, sometimes called a reference angle:

Note that in this case \angle A was used as the reference angle. The side labels would be different if \angle B had been used instead.

Drag the slider to change the size of the reference angle and drag the triangle to change its size.

- What do you notice and wonder about the ratios?
- Can you justify why the ratios stay the same for a given angle, even when you change the size of the triangle?

Right triangles with the same acute angle are similar. Using the proportionality of similar triangles, we can calculate and estimate unknown side lengths and angles in right triangles.

With the given notation of ratios in mind, we then define the following three trigonometric ratios:

That is, for a given reference angle \theta, we have:\sin\theta=\dfrac{\text{opposite}}{\text{hypotenuse}} \qquad \cos\theta=\dfrac{\text{adjacent}}{\text{hypotenuse}} \qquad \tan\theta=\dfrac{\text{opposite}}{\text{adjacent}}

Write the following ratios for the given triangle:

a

\sin\theta

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b

\cos \theta

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c

\tan \theta

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Consider the triangle in the figure. If \sin\theta=\dfrac{4}{5}:

a

Which angle is represented by \theta?

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b

Find the value of \cos\theta.

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c

Find the value of \tan \theta.

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Explain why \sin(x) is the same for any of the triangles in the figure.

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Find the height, HC, of the tree.

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Idea summary

Use the following notation for writing the three trigonometric ratios given the acute reference angle \theta in a right triangle:\sin\theta=\dfrac{\text{opposite}}{\text{hypotenuse}} \qquad \cos\theta=\dfrac{\text{adjacent}}{\text{hypotenuse}} \qquad \tan\theta=\dfrac{\text{opposite}}{\text{adjacent}}

Drag the sliders to change the size of the triangle and measure of the reference angle in the triangle.

- What is the relationship between m \angle ABC and m \angle BAC?
- What do you notice about the trigonometric ratios of the angles?

For the acute angles in a right triangle which are complementary, we can state that:

- The sine of any acute angle is equal to the cosine of its complement: \sin \left( \theta \right) = \cos \left( 90 - \theta \right)
- The cosine of any acute angle is equal to the sine of its complement: \cos \left( \theta \right) = \sin \left( 90 - \theta \right)

Use the diagram to show that \tan \theta = \dfrac{ \sin \theta}{\cos \theta}.

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Consider the following triangle:

a

Write a rule to describe the relationship between \theta and \beta.

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b

Use the rule you found in part (a) to determine the relationship between \sin \theta and \cos \beta.

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Complete the statement: If \cos \left(30 \degree \right) = \dfrac{\sqrt{3}}{2}, then \sin \left( ? \right) = \dfrac{\sqrt{3}}{2}.

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Idea summary

For the acute angles in a right triangle that are complementary, we can state that:

- The sine of any acute angle is equal to the cosine of its complement: \sin \left( \theta \right) = \cos \left( 90 - \theta \right)
- The cosine of any acute angle is equal to the sine of its complement: \cos \left( \theta \right) = \sin \left( 90 - \theta \right)