1. 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.
5.3.1: A Pythagorean Identity

2. 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.
5.3.1: A Pythagorean Identity

3. Key Concepts, continued
5.3.1: A Pythagorean Identity

4. 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 x- and 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

5. 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.
Quadrant II
Angle θ is between 90° and 180°; sin θ is positive and cos θ is negative.
5.3.1: A Pythagorean Identity

6. 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.
5.3.1: A Pythagorean Identity

7. 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.
5.3.1: A Pythagorean Identity

8. Key Concepts, continued
Quadrant
θ in radians
θ in degrees
sin2 θ
cos2 θ
sin2 θ + cos2 θ I
90° 1 II
200°
0.12
0.88
III
270° IV
320°
0.41
0.59
5.3.1: A Pythagorean Identity

9. 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.
5.3.1: A Pythagorean Identity

10. 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
5.3.1: A Pythagorean Identity

11. 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?
5.3.1: A Pythagorean Identity

12. Guided Practice: Example 1, continued Graph y = sin2 θ + cos2 θ.
The graph should appear as follows.
5.3.1: A Pythagorean Identity

13. Guided Practice: Example 1, continued Graph y = 1.
On the same coordinate plane, graph the equation y = 1.
5.3.1: A Pythagorean Identity

14. ✔ Guided Practice: Example 1, continued
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. ✔
5.3.1: A Pythagorean Identity

15. Guided Practice: Example 1, continued
5.3.1: A Pythagorean Identity

16. Guided Practice Example 3
Given that , what is the value of cos A if angle A lies in the first quadrant of the coordinate plane? Round your answer to the nearest hundredth.
5.3.1: A Pythagorean Identity

17. Guided Practice: Example 3, continued
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.
Substitute into the identity equation and solve for cos A.
sin2 A + cos2 A = 1 Pythagorean identity equation
Substitute for sin A.
5.3.1: A Pythagorean Identity

18. Guided Practice: Example 3, continued
Simplify.
Subtract from both sides.
Take the square root of both
sides.
Simplify using a calculator.
cos A is approximately equal to ±0.87.
5.3.1: A Pythagorean Identity

19. ✔ Guided Practice: Example 3, continued
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. ✔
5.3.1: A Pythagorean Identity

20. Guided Practice: Example 3, continued
5.3.1: A Pythagorean Identity