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.
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.
sin(θ)=hypopp
cos(θ)=hypadj
tan(θ)=adjopp
Three other trigonometric functions are cosecant, secant and cotangent. They are defined like this.
csc(θ)=opphyp
sec(θ)=adjhyp
cot(θ)=oppadj
Evaluate the trigonometric functions for the angle θ.
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 |
For a right triangle with legs a and b, and hypotenuse c, the following is true.
a2+b2=c2
For the right triangle, find x and y.
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.
If the value of sine, cosine or tangent 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.
In the same way, inverse cosine (cos-1) and inverse tangent (tan-1) are the inverses of cosine and tangent, respectively.
Determine the measure of the angle θ.
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). Thus, the measure of the angle θ is 27.7∘.