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8. Right Triangles & Trigonometry
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United States of AmericaTennessee
Geometry

8.03 Trigonometric ratios

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Concept summary

A trigonometric ratio, or trigonometric function, is a relatonship between an angle and a pair of side 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:

Right triangle A B C with right angle C. Side A C is labeled Leg adjacent to angle A, side B C labeled Leg opposite angle A, and side A B labeled Hypotenuse.

The adjacent side is the leg of the right triangle that is connected to the angle of reference.

The opposite side is the leg of the right triangle that is across from the angle of reference.

The hypotenuse is the side of the right triangle that is opposite the right angle (the longest side of the right triangle).

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.

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

Sine (sin)

The ratio between the length of the side opposite to a given angle and the hypotenuse of the right triangle

Cosine (cos)

The ratio between the length of the side adjacent to a given angle and the hypotenuse of the right triangle

Tangent (tan)

The ratio between the length of the side opposite and the side adjacent to a given angle.

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}}

Worked examples

Example 1

Write the following ratios for the given triangle:

Right triangle A B C with right angle C. Angle A is labeled alpha, and angle B labeled theta.
a

\sin\theta

Approach

We want to first label the side lengths of the right triangle as shown in the below diagram.

Right Triangle A B C with right angle C. Side A C is labeled Opposite, side B C labeled Adjacent, and side A B labeled Hypotenuse.

The reference angle is \theta, so the opposite side is \overline{AC} as it directly across from the angle of reference, \theta. The hypotenuse is \overline{AB} as it is opposite the right angle. The adjacent side is \overline{BC} as the side is connected to the angle of reference and is not the hypotenuse.

Now that we have the sides of the right triangle labeled, we want to write the correct trigonometric ratio for sine. Sine is the ratio between the side opposite to the given angle and the hypotenuse of the right triangle.

Solution

\sin\theta = \dfrac{AC}{AB}

b

\cos \theta

Approach

We want to reference the sides of the right triangle that we labeled in part (a) to write the correct trigonometric ratio for cosine. Cosine is the ratio between the side adjacent to the given angle and the hypotenuse of a right triangle.

Solution

\cos\theta = \dfrac{BC}{AB}

c

\tan \theta

Approach

We want to reference the sides of the right triangle that we labeled in part (a) to write the correct trigonometric ratio for tangent. Tangent is the ratio between the sides opposite and adjacent to the given angle of a right triangle.

Solution

\tan\theta = \dfrac{AC}{BC}

Example 2

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

Right triangle A B C with right angle A. Side A C has a length of x, A B has a length of 4, and B C has a length of 5.
a

Which angle is represented by \theta?

Approach

In a right triangle, we know that the sine of an angle is equal to the ratio of the side length oposite that angle and the length of the hypotenuse. Since 4 is in the numerator, the side length with length 4 is the opposite side and the hypotenuse has a length of 5. We need to look at the diagram and find the angle that is across from the opposite side with length of 4.

Solution

\angle BCA

Reflection

We could have also named this angle \angle ACB, or even just \angle C since there is no ambiguity as to what that represents.

b

Find the value of \cos\theta.

Approach

From part (a), we labeled the opposite and hypotenuse sides. The adjacent side is labeled x in the diagram. In order to find \cos \theta, we need to find the value of x and then write the trigonometric ratio.

To find the value of x, we can use the Pythagorean theorem.

Solution

\displaystyle a^2+b^2\displaystyle =\displaystyle c^2Pythagorean Theorem
\displaystyle x^2+4^2\displaystyle =\displaystyle 5^2Substitution
\displaystyle x^2+16\displaystyle =\displaystyle 25Simplify
\displaystyle x^2\displaystyle =\displaystyle 9Subtract 16 to both sides
\displaystyle x\displaystyle =\displaystyle 3Square root both sides

Now we want to write the trigonometric ratio for cosine, \dfrac{\text{adjacent}}{\text{hypotenuse}}

\cos\theta = \dfrac{3}{5}

Reflection

We could also realize that this a Pythagorean triple of 3, 4, \text{and } 5 at the start and then write the trigonometric ratio for \cos \theta.

c

Find the value of \tan \theta.

Approach

From part (a), we labeled the opposite and hypotenuse sides. From part (b) we labeled and found the value of the adjacent side. We can use these values to write the tangent ratio, which is the ratio between the sides opposite and adjacent to the given angle of the right triangle.

Solution

\tan\theta = \dfrac{4}{3}

Outcomes

G.N.Q.A.1

Use units as a way to understand real-world problems.*

G.N.Q.A.1.A

Use appropriate quantities in formulas, converting units as necessary.

G.SRT.C.4

Use side ratios in right triangles to define trigonometric ratios.

G.SRT.C.4.A

Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

G.SRT.C.4.B

Explain and use the relationship between the sine and cosine of complementary angles.

G.SRT.C.5

Solve triangles.*

G.SRT.C.5.A

Know and use the Pythagorean Theorem and trigonometric ratios (sine, cosine, tangent, and their inverses) to solve right triangles in a real-world context.

G.MP3

Construct viable arguments and critique the reasoning of others.

G.MP5

Use appropriate tools strategically.

G.MP6

Attend to precision.

G.MP7

Look for and make use of structure.

G.MP8

Look for and express regularity in repeated reasoning.

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