The ratio between the lengths of the opposite side and the hypotenuse in a right triangle for a specific angle, θ, is called the sine of θ.
C
c
The ratio between the lengths of the adjacent side and the hypotenuse in a right triangle for a specific angle, θ, is called the cosine of θ.
A
a
tan(θ)=y/x
B
b
Expression for sin(θ): sin(θ) = y/r
Value when r = 1: sin(θ) = y
C
c
Expression for cos(θ): cos(θ) = x/r
Value when r = 1: cos(θ) = x
Practice makes perfect
a
We want to write an equation representing the relationship between x, y and r. Let's consider the given diagram.
It depicts a right triangle. Notice that x is the length of leg adjacent to angle θ and y is the length of leg opposite angle θ. We can relate those terms using the tangent ratio.
tan(θ)=opposite/adjacent
Let's substitute y for opposite and x for adjacent to obtain the sought relationship.
tan(θ)=y/x
b
Consider the given diagram.
Now, let's recall that in a right triangle, sine of an acute angle is the ratio of lengths of leg opposite that angle and the hypotenuse.
sin( θ) = opposite/hypotenuse
In our case, the lengths of the opposite leg and the hypotenuse are y and r, respectively.
sin( θ) = y/r
Finally, let's consider this expression for r = 1.
sin( θ) = y/1 = y.
Therefore, when r = 1, we have sin( θ) =y.
c
Consider the given diagram.
Now, let's recall that in a right triangle, cosine of an acute angle is the ratio of lengths of leg adjacent to that angle and the hypotenuse.
cos( θ) = adjacent/hypotenuse
In our case, the lengths of the adjacent leg and the hypotenuse are x and r, respectively.
cos( θ) = x/r
Finally, let's consider this expression for r = 1.
cos( θ) = x/1 = x.
Therefore, when r = 1, we have cos( θ) =x.
