Introduction
The Pythagorean Theorem is often used to express the
relationship between known sides of a right triangle and
the triangle’s hypotenuse. The Pythagorean Theorem
can also be written in terms of an angle of a triangle
rather than the triangle’s sides. This form of the equation
can be helpful in solving problems for unknown
information about the sides of a triangle. This lesson
focuses on the equation sin2 θ + cos2 θ = 1, an identity
derived from the Pythagorean Theorem. You will use this
identity to solve various types of problems involving
angles in different quadrants.
1
5.3.1: A Pythagorean Identity
Key Concepts
• An identity is an equation that is true regardless of
what values are chosen for the variables.
• A Pythagorean identity is a trigonometric equation
that is derived from the Pythagorean Theorem. The
primary Pythagorean identity is sin2 θ + cos2 θ = 1.
Other Pythagorean identity equations involving
different trigonometric functions (i.e., tangents,
secants) also exist. Pythagorean identities express
the relationships among the sides and angles of a
right triangle inscribed in a unit circle, as shown on the
following slide.
2
5.3.1: A Pythagorean Identity
Key Concepts, continued
3
5.3.1: A Pythagorean Identity
Key Concepts, continued
• Recall that the Pythagorean Theorem, a2 + b2 = c2, states
that the length of the longest side of a right triangle—the
hypotenuse, c—is equal to the sum of the squares of the
lengths of the other two sides, a and b.
• Also, remember that the center of the unit circle is located
at the origin of the coordinate plane. The unit circle has a
radius of 1, which is also the length of the hypotenuse of a
right triangle drawn in the unit circle. Since c2 = 1, the
Pythagorean Theorem, when applied to a unit circle, can
be written as a2 + b2 = 1.
• Recall that quadrants divide the coordinate plane by an xand y-axis; counterclockwise starting at the top right
quadrant, the four quadrants are labeled I, II, III, and IV.
5.3.1: A Pythagorean Identity
4
Key Concepts, continued
• The following diagrams express the sine and cosine of an
angle θ in a right triangle inscribed within each quadrant of
the unit circle.
Quadrant I
Angle θ is between 0° and 90°;
sin θ and cos θ are both positive.
5.3.1: A Pythagorean Identity
Quadrant II
Angle θ is between 90° and 180°;
sin θ is positive and cos θ is negative.
5
Key Concepts, continued
Quadrant III
Angle θ is between 90° and 270°;
sin θ and cos θ are both negative.
Quadrant IV
Angle θ is between 270° and 360°;
sin θ is negative and cos θ is positive.
6
5.3.1: A Pythagorean Identity
Key Concepts, continued
• The following table shows the values of θ in degrees
and radians for each triangle in the diagrams, as well
as sin2 θ and cos2 θ. The final column of the table
shows the sum of sin2 θ and cos2 θ for each angle
measure.
7
5.3.1: A Pythagorean Identity
Key Concepts, continued
θ in
radians
Quadrant
I
II
III
IV
p
2
sin2 θ
cos2 θ
sin2 θ + cos2 θ
90°
1
0
1
radians
200°
0.12
0.88
1
radians
270°
1
0
1
radians
320°
0.41
0.59
1
radians
10p
9
3p
2
16p
9
θ in
degrees
8
5.3.1: A Pythagorean Identity
Key Concepts, continued
• Notice that all of the values for sin2 θ + cos2 θ are
equal to 1. This indicates that sin2 θ + cos2 θ = 1 is an
identity because the equation remains true for any
value of θ.
9
5.3.1: A Pythagorean Identity
Common Errors/Misconceptions
• misinterpreting variables from a word problem
• miscalculating expressions involving decimals
• using radians instead of degrees and vice versa
• incorrectly choosing the reference angle
10
5.3.1: A Pythagorean Identity
Guided Practice
Example 1
Use a graphing calculator to graph y = sin2 θ + cos2 θ
and y = 1 on the same graph. What do you observe
about these two graphs?
11
5.3.1: A Pythagorean Identity
Guided Practice: Example 1, continued
1. Graph y = sin2 θ + cos2 θ.
The graph should appear as follows.
12
5.3.1: A Pythagorean Identity
Guided Practice: Example 1, continued
2. Graph y = 1.
On the same coordinate plane, graph the equation y = 1.
13
5.3.1: A Pythagorean Identity
Guided Practice: Example 1, continued
3. Make an observation about the graphs of
y = sin2 θ + cos2 θ and y = 1.
It can be seen from the graph that the two graphs are
the same. Therefore, the equations are equal:
y = 1 = sin2 θ + cos2 θ.
✔
14
5.3.1: A Pythagorean Identity
Guided Practice: Example 1, continued
15
5.3.1: A Pythagorean Identity
Guided Practice
Example 3
1
Given that sin A = , what is the value of cos A if angle A
2
lies in the first quadrant of the coordinate plane? Round
your answer to the nearest hundredth.
16
5.3.1: A Pythagorean Identity
Guided Practice: Example 3, continued
1. Use the Pythagorean identity sin2 θ +
cos2 θ = 1 to determine the value of cos A.
For this problem, replace θ with A in the identity
equation: sin2 A + cos2 A = 1.
1
Substitute sin A = into the identity equation and
2
solve for cos A.
sin2 A + cos2 A = 1
2
æ 1ö
equation
2
+
cos
A=1
çè 2 ÷ø
5.3.1: A Pythagorean Identity
Pythagorean identity
1
Substitute
2
for sin A.
17
Guided Practice: Example 3, continued
1
4
+ cos2 A = 1
cos A =
2
cos A = ±
3
4
3
4
cos A » ±0.87
Simplify.
Subtract
1
from both sides.
4
Take the square root of both
sides.
Simplify using a calculator.
cos A is approximately equal to ±0.87.
18
5.3.1: A Pythagorean Identity
Guided Practice: Example 3, continued
2. Determine the value of cos A if angle A lies
in the first quadrant.
In the first quadrant, cosine is positive, so use the
positive square root for the result. Therefore, cos A is
approximately equal to 0.87.
✔
19
5.3.1: A Pythagorean Identity
Guided Practice: Example 3, continued
20
5.3.1: A Pythagorean Identity