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Using Right Triangle Trigonometry

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

sin(θ)=opphyp\sin(\theta)=\dfrac{\text{opp}}{\text{hyp}}

cos(θ)=adjhyp\cos(\theta)=\dfrac{\text{adj}}{\text{hyp}}

tan(θ)=oppadj\tan(\theta)=\dfrac{\text{opp}}{\text{adj}}




Three other trigonometric functions are cosecant, secant and cotangent. They are defined like this.

csc(θ)=hypopp\csc(\theta)=\dfrac{\text{hyp}}{\text{opp}}

sec(θ)=hypadj\sec(\theta)=\dfrac{\text{hyp}}{\text{adj}}

cot(θ)=adjopp\cot(\theta)=\dfrac{\text{adj}}{\text{opp}}




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

Evaluate the trigonometric functions for the angle θ.\theta.

Show Solution
Solution

To begin, we should label the triangle's sides. The longest side is the hypotenuse, the side next to θ\theta is the adjacent side and the side across from θ\theta 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)\sin(\theta) opphyp\dfrac{\text{opp}}{\text{hyp}} 35=0.6\dfrac{3}{5}=0.6
cos(θ)\cos(\theta) adjhyp\dfrac{\text{adj}}{\text{hyp}} 45=0.8\dfrac{4}{5}=0.8
tan(θ)\tan(\theta) oppadj\dfrac{\text{opp}}{\text{adj}} 34=0.75\dfrac{3}{4}=0.75
csc(θ)\csc(\theta) hypopp\dfrac{\text{hyp}}{\text{opp}} 531.67\dfrac{5}{3}\approx 1.67
sec(θ)\sec(\theta) hypadj\dfrac{\text{hyp}}{\text{adj}} 54=1.25\dfrac{5}{4}=1.25
cot(θ)\cot(\theta) adjopp\dfrac{\text{adj}}{\text{opp}} 431.33\dfrac{4}{3}\approx 1.33
Rule

Pythagorean Theorem

For a right triangle with legs aa and b,b, and hypotenuse c,c, the following is true.

a2+b2=c2a^2+b^2=c^2

This can be proven using area.
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Exercise

For the right triangle, find xx and y.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.x. Since the adjacent side is given, we must use cosθ\cos\theta to determine x.x.
cos(θ)=adjhyp\cos(\theta) =\dfrac{\text{adj}}{\text{hyp}}
cos(30)=4hyp\cos({\color{#0000FF}{30^\circ}}) =\dfrac{{\color{#009600}{4}}}{\text{hyp}}
cos(30)=4x\cos(30^\circ) =\dfrac{4}{{\color{#0000FF}{x}}}
xcos(30)=4x\cdot \cos(30^\circ) = 4
x=4cos(30)x = \dfrac{4}{\cos(30^\circ)}
x=4.61880x = 4.61880\ldots
x4.62x \approx 4.62
The length of the hypotenuse is approximately 4.62.4.62. We now know the lengths of two sides in the right triangle. The third side, y,y, can be calculated in the same way using tanθ,\tan\theta, or by using the Pythagorean Theorem. We'll use the latter. Remember that cc is the hypotenuse. The values of aa and bb can be assigned arbitrarily.
a2+b2=c2a^2+b^2=c^2
42+b2=(4.61880)2{\color{#0000FF}{4}}^2+b^2=({\color{#009600}{4.61880\ldots}})^2
42+y2=(4.61880)24^2+{\color{#0000FF}{y}}^2=(4.61880\ldots)^2
y2=(4.61880)242y^2=(4.61880\ldots)^2-4^2
y2=5.33333y^2=5.33333\ldots
y=±2.30940y=\pm 2.30940\ldots
y>0 y \gt 0
y=2.30940y= 2.30940\ldots
y2.31y\approx 2.31
Thus, the two unknown sides in the right triangle have the lengths x4.62andy2.31. x\approx4.62 \quad \text{and} \quad y\approx 2.31.
Concept

Inverse Trigonometric Functions

If the value of sine, cosine or tangent for a specific angle, θ\theta, 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.\sin^{\text{-} 1}.

In the same way, inverse cosine (cos-1\cos^{\text{-} 1}) and inverse tangent (tan-1\tan^{\text{-} 1}) are the inverses of cosine and tangent, respectively.

InverseTrigFuncs 1.svg
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Exercise

Determine the measure of the angle θ.\theta.

Show Solution
Solution

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

Since the hypotenuse and the adjacent side are given, we'll use cosθ.\cos\theta.

cos(θ)=adjhyp\cos(\theta)=\dfrac{\text{adj}}{\text{hyp}}
cos(θ)=5.46.1\cos(\theta)=\dfrac{{\color{#0000FF}{5.4}}}{{\color{#009600}{6.1}}}
Since, cos(θ)=5.46.1,\cos(\theta)=\dfrac{5.4}{6.1}, it follows that θ=cos-1(5.46.1).\theta=\cos^{\text{-} 1} \left( \dfrac{5.4}{6.1} \right).
θ=cos-1(5.46.1)\theta=\cos^{\text{-} 1} \left( \dfrac{5.4}{6.1} \right)
θ=27.71820\theta=27.71820\ldots
θ27.7\theta\approx 27.7
Thus, the measure of the angle θ\theta is 27.7.27.7^\circ.
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