Start by identifying the hypotenuse of the right triangle. Then find the sides that are opposite and adjacent to ∠ A.
lsin A=sqrt(5)/5
lcos A=2sqrt(5)/5
ltan A=1/2
Practice makes perfect
For the given right triangle, we want to write the ratios for the sine, cosine, and tangent of ∠ A. Let's start by identifying the hypotenuse of the triangle and the sides that are opposite and adjacent to ∠ A.
We see that the length of the hypotenuse is yet unknown, let's call it x. The length of the side adjacent to ∠ A is 12 and the length of the side opposite to ∠ A is 6. We can find x by using the Pythagorean Theorem. Let's write an equation according to this theorem.
6^2 + 12^2 = x^2
Now we can solve this equation to find x.
Since the length of the side of the triangle cannot be negative, we only consider the positive solution. Therefore, the length of the hypotenuse is 6sqrt(5).
With this information, we can find the desired ratios.
Remember to rationalize denominators that contain square roots.
Ratio
Definition
Value
sin A
Length of leg opposite∠ A/Length of hypotenuse
6/6sqrt(5) = sqrt(5)/5
cos A
Length of leg adjacent∠ A/Length of hypotenuse
12/6sqrt(5) = 2sqrt(5)/5
tan A
Length of leg opposite∠ A/Length of leg adjacent∠ A
6/12 = 1/2
Showing Our Work
Rationalizing a Denominator
To rationalize a denominator means to eliminate any irrational number in the denominator. To do so, we expand the fraction by multiplying both the numerator and denominator by the irrational number. Let's rationalize the denominator of 66sqrt(5).
