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{{ printedBook.courseTrack.name }} {{ printedBook.name }} Suppose $△ABC$ and $△PQR$ 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, $PRAC =PQAB ⇔ABAC =PQPR .$
The second proportion yields *trigonometric ratios* — ratios that relate the side lengths of one right triangle. Namely, the *tangent of $B$* is defined as $ABAC .$
$tan(B)=ABAC $

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

As drawn, $BC$ is the hypotenuse of $△ABC.$ The remaining sides can be named relative to the marked angle $θ.$ Because $AB$ is next to $∠θ,$ it is called the *adjacent* side. Similarly, because $AC$ lies across from $∠θ,$ it is called the *opposite side.*

Sine, cosine, and tangent of $θ$ are defined as follows. $sin(θ)=hypopp cos(θ)=hypadj tan(θ)=adjopp $

Trigonometric ratios can be used to determine unknown side lengths in right triangles.Determine sine, cosine, and tangent of the angle $θ.$

Show Solution

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

$sin(θ)=hypopp $

$sin(θ)=9.77.2 $

UseCalcUse a calculator

$sin(θ)=0.74226…$

RoundDecRound to ${\textstyle 2 \, \ifnumequal{2}{1}{\text{decimal}}{\text{decimals}}}$

$sin(θ)≈0.74$

ratio | definition | value |
---|---|---|

$sin(θ)$ | $hypopp $ | $9.77.2 ≈0.74$ |

$cos(θ)$ | $hypadj $ | $9.76.5 ≈0.67$ |

$tan(θ)$ | $adjopp $ | $6.57.2 ≈1.11$ |

By the Interior Angles Theorem, the sum of the interior angle measures of a triangle is $180_{∘}.$ For a right triangle, since one angle measures $90_{∘},$ the other two angles are acute.

It follows that the sum of the measures of $∠A$ and $∠B$ is $90_{∘}.$ Therefore, they are complementary angles. $m∠A+m∠B=90_{∘}$ The sine and cosine ratios of complementary angles have a special relationship. To explore this, the three sides in the triangle will be labeled $x,$ $y,$ and $z.$

The definitions of sine and cosine can be applied as follows. $sin(A)=yz sin(B)=yx cos(A)=yx cos(B)=yz $ It can be seen that $sin(A)=cos(B)$ and $cos(A)=sin(B).$ This is true for all pairs of complementary angles. If an acute angle is named $θ,$ its complementary angle can be written $(90_{∘}−θ).$ Thus, the relationship can be written as follows.

$sin(θ)=cos(90_{∘}−θ)cos(θ)=sin(90_{∘}−θ)$ {{ 'mldesktop-placeholder-grade' | message }} {{ article.displayTitle }}!

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