b Compare the values of sine and cosine within each pair. What can you tell about the sum of the angle measures of each pair?
C
c What is the sum of the measures of two angles that are not right angles in any right triangle? Find the sine and cosine ratios corresponding to them.
Second Pair: cos(25^(∘))≈ 0.9063, sin(65^(∘))≈ 0.9063
B
bValues of Sine and Cosine: They are the same.
Angles: They are complementary angles.
C
c See solution.
Practice makes perfect
a We are asked to find the values of each pair of expressions. To do this, we will use a calculator.
First Pair (i)
Second Pair (ii)
sin(80^(∘))≈ 0.9848 cos(10^(∘))≈ 0.9848
cos(25^(∘))≈ 0.9063 sin(65^(∘))≈ 0.9063
b Examining the values from Part A, we can notice that the values of sine and cosine in each pair are the same.
First Pair (i)
Second Pair (ii)
sin(80^(∘))≈ 0.9848 cos(10^(∘))≈ 0.9848
cos(25^(∘))≈ 0.9063 sin(65^(∘))≈ 0.9063
Moreover, the sum of the angle measures in each pair is equal to 90^(∘).
First Pair (i)
Second Pair (ii)
sin( 80^(∘))≈ 0.9848 cos( 10^(∘))≈ 0.9848
cos( 25^(∘))≈ 0.9063 sin( 65^(∘))≈ 0.9063
80^(∘)+ 10^(∘)=90^(∘)
25^(∘)+ 65^(∘)=90^(∘)
This allows us to conclude that the angles in each pair are complementary angles.
c Let's begin by recalling the definitions of sine and cosine.
Term
Definition
Ratio
Sine
The ratio between the lengths of the opposite side and the hypotenuse in a right triangle for a specific angle.
sin(θ)=opposite/hypotenuse
Cosine
The ratio between the lengths of the adjacent side and the hypotenuse in a right triangle for a specific angle.
cos(θ)=adjacent/hypotenuse
Next, we will consider an arbitrary right triangle △ ABC.
By the Triangle Angle Sum Theorem, the sum of the measures of the angles of a triangle is 180^(∘).
m∠ A+m∠ B+m∠ C=180^(∘)
Since m∠ C is a right angle, we can substitute its measure with 90^(∘).
m∠ A+m∠ B+ 90^(∘)=180^(∘)
⇓
m∠ A+m∠ B=90^(∘)
Therefore, no matter what measures ∠ A and ∠ B have, they will always be complementary angles. Let's find and analyze the ratios for sin A and cos B.
As we can see, the ratios of sine and cosine for these angles are the same. Therefore, the following relationship between the sine and cosine of two complementary angles is true.
sin A=cos B
This explains why the results from Part A make sense.
