Compare the sine and cosine of the acute angles of a general right triangle and notice that these angles are complementary.
Relationship: The sine of one angle is equal to the cosine of the complementary angle. Explanation: See solution.
cos(50^(∘)) = 0.64
Practice makes perfect
Let's consider the right triangle below. Notice that ∠ Q and ∠ R are complementary angles.
In the following table, we will find the sine and cosine of the acute angles.
Angle
Sine
Cosine
∠ Q
sin Q = PR/RQ
cos Q = PQ/RQ
∠ R
sin R = PQ/RQ
cos R= PR/RQ
From the table above, we can see the following relation between the sine and cosine of complementary angles.
sin( ∠ Q)=cos( ∠ R)
and
sin( ∠ R) = cos( ∠ Q)
In conclusion, the sine of one angle is equal to the cosine of the complementary angle.
Finding cos(50^(∘))
From the relationship we found before, we have the following equation.
cos(50^(∘)) = sin(complementary of50^(∘))
The complementary angle of 50^(∘) is an angle with measure 90^(∘) -50^(∘), which is a 40^(∘) angle.
cos(50^(∘)) = sin(40^(∘))
Finally, remember that we are told that sin(40^(∘)) = 0.64, and therefore, we have that cos(50^(∘)) = 0.64.
