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Defining and Using Trigonometric Ratios


Similar Right Triangles

Suppose ABC\triangle ABC and PQR\triangle PQR are right triangles and share another congruent angle.

By the Angle-Angle Similarity Theorem, these triangles are similar. As a result, corresponding sides are proportional. In fact, the ratio between corresponding sides is constant. For example, ACPR=ABPQACAB=PRPQ. \dfrac{AC}{PR} = \dfrac{AB}{PQ}\quad \Leftrightarrow \quad \dfrac{AC}{AB}=\dfrac{PR}{PQ}. The second proportion yields trigonometric ratios — ratios that relate the side lengths of one right triangle. Namely, the tangent of BB is defined as ACAB.\frac{AC}{AB}. tan(B)=ACAB \tan(B)=\dfrac{AC}{AB}

Because proportions of all corresponding sides can be written, there are trigonometric ratios that relate all pairs of the three sides of a triangle.

Trigonometric Ratios

A trigonometric ratio relates two side lengths of a right triangle. The three most notable are sine, cosine, and tangent. Consider the right triangle ABC.\triangle ABC.

As drawn, BC\overline{BC} is the hypotenuse of ABC.\triangle ABC. The remaining sides can be named relative to the marked angle θ.\theta. Because AB\overline{AB} is next to θ,\angle \theta, it is called the adjacent side. Similarly, because AC\overline{AC} lies across from θ,\angle \theta, it is called the opposite side.

Sine, cosine, and tangent of θ\theta are defined as follows. sin(θ)=opphypcos(θ)=adjhyptan(θ)=oppadj \sin(\theta)=\dfrac{\text{opp}}{\text{hyp}} \quad \cos(\theta)=\dfrac{\text{adj}}{\text{hyp}} \quad \tan(\theta)=\dfrac{\text{opp}}{\text{adj}}

Trigonometric ratios can be used to determine unknown side lengths in right triangles.

Determine sine, cosine, and tangent of the angle θ.\theta.

Show Solution

To begin we need to determine the triangle's sides relative to θ.\theta. The side across from the right angle is always the hypotenuse. The side next to θ\theta is the adjacent side and the side across it is the opposite side. Thus, opposite: 7.2adjacent: 6.5andhypotenuse: 9.7. \text{opposite: } 7.2 \quad \text{adjacent: } 6.5 \quad \text{and} \quad \text{hypotenuse: } 9.7. Now that the sides are known, we can use the definition of sine to determine its value.

sin(θ)0.74\sin(\theta)\approx 0.74
Thus, sin(θ)0.74.\sin(\theta)\approx 0.74. We can find the values of cosine and tangent in the same way.
ratio definition value
sin(θ)\sin(\theta) opphyp\dfrac{\text{opp}}{\text{hyp}}\dfrac{7.2}{9.7}\approx 0.74
cos(θ)\cos(\theta) adjhyp\dfrac{\text{adj}}{\text{hyp}}\dfrac{6.5}{9.7}\approx 0.67
tan(θ)\tan(\theta) oppadj\dfrac{\text{opp}}{\text{adj}}\dfrac{7.2}{6.5}\approx 1.11


Sine and Cosine of Complementary Angles

By the Interior Angles Theorem, the sum of the interior angle measures of a triangle is 180.180^\circ. For a right triangle, since one angle measures 90,90^\circ, the other two angles are acute.

It follows that the sum of the measures of A\angle A and B\angle B is 90.90^\circ. Therefore, they are complementary angles. mA+mB=90 m\angle A + m\angle B =90^\circ The sine and cosine ratios of complementary angles have a special relationship. To explore this, the three sides in the triangle will be labeled x,x, y,y, and z.z.

The definitions of sine and cosine can be applied as follows. sin(A)=zycos(A)=xysin(B)=xycos(B)=zy\begin{aligned} \sin(A)=\dfrac{z}{y} & \quad & \cos(A)=\dfrac{x}{y} \\ \\ \sin(B)=\dfrac{x}{y} & \quad & \cos(B)=\dfrac{z}{y} \end{aligned} It can be seen that sin(A)=cos(B)\sin(A)=\cos(B) and cos(A)=sin(B).\cos(A)=\sin(B). This is true for all pairs of complementary angles. If an acute angle is named θ,\theta, its complementary angle can be written (90θ).(90^\circ - \theta). Thus, the relationship can be written as follows.

sin(θ)=cos(90θ)cos(θ)=sin(90θ) \sin(\theta)=\cos(90^\circ -\theta) \qquad \cos(\theta)=\sin(90^\circ -\theta)
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