Start by identifying the hypotenuse of the right triangle. Then find the sides that are opposite and adjacent to ∠ M.
sin M=sqrt(3)/2,
cos M=1/2,
tan M=sqrt(3)
Practice makes perfect
For the given right triangle, we want to write the ratios for the sine, cosine, and tangent of ∠ M.
Let's start by identifying the hypotenuse of the triangle and the sides that are opposite and adjacent to ∠ M.
We see that the length of the hypotenuse is 4. The length of the side adjacent to ∠ M is 2 and the length of the side opposite to ∠ M is 2sqrt(3). With this information, we can find the desired ratios.
Ratio
Definition
Value
sin M
Length of leg opposite∠ M/Length of hypotenuse
2sqrt(3)/4=sqrt(3)/2
cos M
Length of leg adjacent∠ M/Length of hypotenuse
2/4=1/2
tan M
Length of leg opposite∠ M/Length of leg adjacent∠ M
2sqrt(3)/2=sqrt(3)
Extra
Reciprocal Ratios
Later in the book we will learn that the sine, cosine, and tangent ratios each have a reciprocal ratio. The reciprocal ratios are cosecant, secant, and cotangent. These terms are often shortened to csc, sec, and cot, respectively.
csc θ = hypotenuse/opposite, sec θ = hypotenuse/adjacent
cot θ = adjacent/opposite
In the given right triangle, the length of the hypotenuse is 4. The length of the side adjacent to ∠ M is 2 and the length of the side opposite to ∠ M is 2 sqrt(3). With this information, we can also find the reciprocal ratios of the ratios that we found in the exercise.
Ratio
Definition
Value
csc M
Length of hypotenuse/Length of leg opposite∠ M
4/2sqrt(3)=2sqrt(3)/3
sec M
Length of hypotenuse/Length of leg adjacent∠ M
4/2=2
cot M
Length of leg adjacent∠ M/Length of leg opposite∠ M
