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Trigonometry is the study of the relationships between the angles and sides in triangles. These relationships can be used to calculate different characteristics of triangles.

In a right triangle, the hypotenuse is always the longest side of the triangle and lies across from the right angle. The triangle's other two sides are called legs. If one of the acute angles in the triangle is labeled, the legs can be discussed relative to that angle. If $θ$ is labeled as shown, the side that forms the angle is the *adjacent* side and the side not touching the angle is the *opposite* side.

Change angles

Trigonometric Functions | ||
---|---|---|

$sinθ=hypopp $ | $cosθ=hypadj $ | $tanθ=adjopp $ |

Three other trigonometric functions are cosecant, secant, and cotangent. These are called reciprocal trigonometric functions. It is worth mentioning that these functions must not be confused with the inverse trigonometric functions, which are defined entirely differently.

Reciprocal Trigonometric Functions | ||
---|---|---|

$cscθ=opphyp $ | $secθ=adjhyp $ | $cotθ=oppadj $ |

The trigonometric functions only state a ratio between two sides in a right triangle; they do not give the lengths of the respective sides. However, if one acute angle and one side are known, the lengths of the other sides can be determined.

Evaluate the trigonometric functions for the angle $θ.$

Show Solution

To begin, we should label the triangle's sides. The longest side is the hypotenuse, the side next to $θ$ is the adjacent side and the side across from $θ$ is the opposite side.

We can now evaluate the trigonometric functions by using the definitions and the lengths of the sides in the triangle.

trig function | definition | value |
---|---|---|

$sin(θ)$ | $hypopp $ | $53 =0.6$ |

$cos(θ)$ | $hypadj $ | $54 =0.8$ |

$tan(θ)$ | $adjopp $ | $43 =0.75$ |

$csc(θ)$ | $opphyp $ | $35 ≈1.67$ |

$sec(θ)$ | $adjhyp $ | $45 =1.25$ |

$cot(θ)$ | $oppadj $ | $34 ≈1.33$ |

$a_{2}+b_{2}=c_{2}$

This can be proven using area.

For the right triangle, find $x$ and $y.$

Show Solution

To begin, we should note that since the given triangle is a right triangle, both the trigonometric ratios and the Pythagorean Theorem might be useful tools. The given information includes one angle and one side. Since two side lengths must be known to use the Pythagorean Theorem, we'll have to use a trigonometric ratio. We can label the sides of the triangle relative to the labeled angle.

Let's first find $x.$ Since the adjacent side is given, we must use $cosθ$ to determine $x.$$cos(θ)=hypadj $

SubstituteII

$θ=30_{∘}$, $adj=4$

$cos(30_{∘})=hyp4 $

Substitute

$hyp=x$

$cos(30_{∘})=x4 $

MultEqn

$LHS⋅x=RHS⋅x$

$x⋅cos(30_{∘})=4$

DivEqn

$LHS/cos(30_{∘})=RHS/cos(30_{∘})$

$x=cos(30_{∘})4 $

UseCalc

Use a calculator

$x=4.61880…$

RoundDec

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

$x≈4.62$

$a_{2}+b_{2}=c_{2}$

SubstituteII

$a=4$, $c=4.61880…$

$4_{2}+b_{2}=(4.61880…)_{2}$

Substitute

$b=y$

$4_{2}+y_{2}=(4.61880…)_{2}$

SubEqn

$LHS−4_{2}=RHS−4_{2}$

$y_{2}=(4.61880…)_{2}−4_{2}$

UseCalc

Use a calculator

$y_{2}=5.33333…$

SqrtEqn

$LHS =RHS $

$y=±2.30940…$

$y>0$

$y=2.30940…$

RoundDec

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

$y≈2.31$

If the value of a trigonometric ratio for a specific angle $θ$ is known, it is possible to calculate the measure of the angle using inverse trigonometric functions. The inverse function of sine is called inverse sine

and is written as $sin_{-1}.$

The rest of the inverse trigonometric functions are defined in a similar way.

Function | Inverse Function |
---|---|

$y=sinθ$ | $θ=sin_{-1}y$ |

$y=cosθ$ | $θ=cos_{-1}y$ |

$y=tanθ$ | $θ=tan_{-1}y$ |

$y=cotθ$ | $θ=cot_{-1}y$ |

$y=secθ$ | $θ=sec_{-1}y$ |

$y=cscθ$ | $θ=csc_{-1}y$ |

As long as the appropriate sides are being used, the same angle can be found using different inverse trigonometric functions.

Determine the measure of the angle $θ.$

Show Solution

In order to solve for $θ,$ we must use inverse trigonometric ratios. To begin, we'll label the sides of the triangle relative to $θ.$

Since the hypotenuse and the adjacent side are given, we'll use $cosθ.$

Since, $cos(θ)=6.15.4 ,$ it follows that $θ=cos_{-1}(6.15.4 ).$$θ=cos_{-1}(6.15.4 )$

UseCalc

Use a calculator

$θ=27.71820…$

RoundDec

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

$θ≈27.7$

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