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.

Concept

Sides in a Right Triangle

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

Note that the opposite and adjacent sides change when

θ

changes, but the hypotenuse is always the same.

Concept

Trigonometric Functions

By using the sides of a right triangle, six trigonometric functions of the acute angle

θ

can be defined. The most common of these are sine, cosine, and tangent.

Trigonometric Functions
$sin θ = \frac{opp}{hyp}$	$cos θ = \frac{adj}{hyp}$	$tan θ = \frac{opp}{adj}$

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
$csc θ = \frac{hyp}{opp}$	$sec θ = \frac{hyp}{adj}$	$cot θ = \frac{adj}{opp}$

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

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Exercise

Evaluate the trigonometric functions for the angle $θ .$

Show Solution

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
$sin (θ)$	$\frac{opp}{hyp}$	$\frac{3}{5} = 0.6$
$cos (θ)$	$\frac{adj}{hyp}$	$\frac{4}{5} = 0.8$
$tan (θ)$	$\frac{opp}{adj}$	$\frac{3}{4} = 0.75$
$csc (θ)$	$\frac{hyp}{opp}$	$\frac{5}{3} \approx 1.67$
$sec (θ)$	$\frac{hyp}{adj}$	$\frac{5}{4} = 1.25$
$cot (θ)$	$\frac{adj}{opp}$	$\frac{4}{3} \approx 1.33$

Rule

Pythagorean Theorem

For a right triangle with legs

a

and

b,

and hypotenuse

c,

the following is true.

$a^{2} + b^{2} = c^{2}$

This can be proven using area.

Find the side lengths of the right triangle

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Exercise

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

Show Solution

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

cos θ

to determine

x .

cos (θ) = \frac{adj}{hyp}

SubstituteII

$θ = 3 0^{\circ}$ , $adj = 4$

cos (3 0^{\circ}) = \frac{4}{hyp}

Substitute

$hyp = x$

cos (3 0^{\circ}) = \frac{4}{x}

MultEqn

$LHS \cdot x = RHS \cdot x$

x \cdot cos (3 0^{\circ}) = 4

DivEqn

$LHS / cos (3 0^{\circ}) = RHS / cos (3 0^{\circ})$

x = \frac{4}{cos ( 3 0 ^{\circ} )}

UseCalc

Use a calculator

x = 4.61880 \dots

RoundDec

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

x \approx 4.62

The length of the hypotenuse is approximately

4.62 .

We now know the lengths of two sides in the right triangle. The third side,

y,

can be calculated in the same way using

tan θ,

or by using the Pythagorean Theorem. We'll use the latter. Remember that

c

is the hypotenuse. The values of

a

and

b

can be assigned arbitrarily.

a^{2} + b^{2} = c^{2}

SubstituteII

$a = 4$ , $c = 4.61880 \dots$

4^{2} + b^{2} = (4.61880 \dots)^{2}

Substitute

$b = y$

4^{2} + y^{2} = (4.61880 \dots)^{2}

SubEqn

$LHS - 4^{2} = RHS - 4^{2}$

y^{2} = (4.61880 \dots)^{2} - 4^{2}

UseCalc

Use a calculator

y^{2} = 5.33333 \dots

SqrtEqn

$LHS = RHS$

y = \pm 2.30940 \dots

$y > 0$

y = 2.30940 \dots

RoundDec

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

y \approx 2.31

Thus, the two unknown sides in the right triangle have the lengths

x \approx 4.62 and y \approx 2.31 .

Concept

Inverse Trigonometric Functions

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
$y = sin θ$	$θ = sin^{- 1} y$
$y = cos θ$	$θ = cos^{- 1} y$
$y = tan θ$	$θ = tan^{- 1} y$
$y = cot θ$	$θ = cot^{- 1} y$
$y = sec θ$	$θ = sec^{- 1} y$
$y = csc θ$	$θ = csc^{- 1} y$

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

Determine the angle

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Exercise

Determine the measure of the angle $θ .$

Show Solution

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 $cos θ .$

cos (θ) = \frac{adj}{hyp}

SubstituteII

$adj = 5.4$ , $hyp = 6.1$

cos (θ) = \frac{5 . 4}{6 . 1}

Since,

cos (θ) = \frac{5 . 4}{6 . 1},

it follows that

θ = cos^{- 1} (\frac{5 . 4}{6 . 1}) .

θ = cos^{- 1} (\frac{5 . 4}{6 . 1})

UseCalc

Use a calculator

θ = 27.71820 \dots

RoundDec

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

θ \approx 27.7

Thus, the measure of the angle

θ

is

27 . 7^{\circ} .

Using Right Triangle Trigonometry

Concept

Sides in a Right Triangle

Concept

Trigonometric Functions

Example

Evaluate the trigonometric functions

Rule

Pythagorean Theorem

Example

Find the side lengths of the right triangle

Concept

Inverse Trigonometric Functions

Example

Determine the angle