Trigonometric
Ratios
and Complementary
Trigonometric
Ratios
and
Angles
Complementary Angles
8-2-EXT
8-2-EXT
Lesson Presentation
HoltMcDougal
GeometryGeometry
Holt
8-2-EXT
Trigonometric Ratios and Complementary
Angles
Objectives
Use the relationship between the sine
and cosine of complementary angles.
Holt McDougal Geometry
8-2-EXT
Trigonometric Ratios and Complementary
Angles
Vocabulary
cofunction
Holt McDougal Geometry
8-2-EXT
Trigonometric Ratios and Complementary
Angles
The acute angles of a right triangle are
complementary angles. If the measure of one of the
two acute angles is given, the measure of the second
acute angle can be found by subtracting the given
measure from 90°.
Holt McDougal Geometry
Trigonometric Ratios and Complementary
Angles
Example 1: Finding the Sine and Cosine of Acute
Angles
8-2-EXT
Find the sine and cosine of
the acute angles in the
right triangle shown.
Start with the sine and cosine of ∠A.
opposite
sin A =
hypotenuse
Holt McDougal Geometry
12
=
37
8-2-EXT
Trigonometric Ratios and Complementary
Angles
Example 1: Continue
adjacent
cos A =
hypotenuse
=
35
37
Then, find the sine and cosine of ∠B.
opposite
sin B =
hypotenuse
adjacent
cos B =
hypotenuse
Holt McDougal Geometry
35
=
37
=
12
37
8-2-EXT
Trigonometric Ratios and Complementary
Angles
Check It Out! Example 1
Find the sine and cosine of the acute angles of
a right triangle with sides 10, 24, 26. (Use A
for the angle opposite the side with length 10
and B for the angle opposite the side with
length 24.)
sin A =
opposite
hypotenuse
adjacent
cos A =
hypotenuse
Holt McDougal Geometry
=
10
5
=
26
13
12
24
=
=
13
26
8-2-EXT
Trigonometric Ratios and Complementary
Angles
Check It Out! Example 1 Continued
sin B =
opposite
hypotenuse
adjacent
Cos B =
hypotenuse
Holt McDougal Geometry
24
12
=
=
26
13
5
10
=
=
13
26
8-2-EXT
Trigonometric Ratios and Complementary
Angles
The trigonometric function of the complement of
an angle is called a cofunction. The sine and
cosines are cofunctions of each other.
Holt McDougal Geometry
8-2-EXT
Trigonometric Ratios and Complementary
Angles
Example 2: Writing Sine in Cosine Terms and Cosine
in Sine Terms
A. Write sin 52° in terms of the cosine.
sin 52° = cos(90 – 52)°
= cos 38
B. Write cos 71° in terms of the sine.
cos 71° = sin(90 – 71)°
= sin 19
Holt McDougal Geometry
8-2-EXT
Trigonometric Ratios and Complementary
Angles
Check It Out! Example 2
A. Write sin 28° in terms of the cosine.
sin 28° = cos(90 – 28)°
= cos 62
B. Write cos 51° in terms of the sine.
cos 51° = sin(90 – 51)°
= sin 39
Holt McDougal Geometry
8-2-EXT
Trigonometric Ratios and Complementary
Angles
Example 3: Finding Unknown Angles
Find the two angles that satisfy the equation
below.
sin (x + 5)° = cos (4x + 10)°
If sin (x + 5)° = cos (4x + 10)° then (x + 5)° and
(4x + 10)° are the measures of complementary
angles. The sum of the measures must be 90°.
x + 5 + 4x + 10 = 90
5x + 15 = 90
5x = 75
x = 15
Holt McDougal Geometry
8-2-EXT
Trigonometric Ratios and Complementary
Angles
Example 3: Continued
Substitute the value of x into the original expression
to find the angle measures.
The measurements of the two angles are 20° and 70°.
Holt McDougal Geometry
8-2-EXT
Trigonometric Ratios and Complementary
Angles
Check It Out! Example 3
Find the two angles that satisfy the equation
below.
A. sin(3x + 2)° = cos(x + 44)°
If sin(3x + 2)° = cos(x + 44)° then (3x + 2)° and
(x + 44)° are the measures of complementary
angles. The sum of the measures must be 90°.
3x + 2 + x + 44 = 90
4x + 46 = 90
4x = 44
x = 11
Holt McDougal Geometry
8-2-EXT
Trigonometric Ratios and Complementary
Angles
Check It Out! Example 3 Continued
Substitute the value of x into the original
expression to find the angle measures.
The measurements of the two angles are 35° and 55°.
B. sin(2x + 20)° = cos(3x + 30)°
If sin(2x + 20)° = cos(3x + 30)° then (2x + 20)°
and (3x + 30)° are the measures of
complementary angles. The sum of the measures
must be 90°.
Holt McDougal Geometry
8-2-EXT
Trigonometric Ratios and Complementary
Angles
Check It Out! Example 3 Continued
2x + 20 + 3x + 30 = 90
5x + 50 = 90
5x = 40
x=8
Substitute the value of x into the original expression
to find the angle measures.
The measurements of the two angles are 36° and 54°.
Holt McDougal Geometry