Download 17.3 Using a Pythagorean Identity Date: ______ Explore 1. Proving

H. Algebra 2
17.3 Notes
17.3 Using a Pythagorean Identity
Date: ___________
Explore 1. Proving a Pythagorean Identity
Learning Target F: I can prove the Pythagorean Identity.
In the previous lesson, you learned that the coordinates of any point (x, y) that lies on the unit circle
𝑦
where the terminal ray of an angle θ intersects the circle are x = cos θ and y = sin θ, and that tan θ = 𝑥 .
sin 𝜃
Combining these facts gives the identity tan 𝜃 = cos 𝜃, which is true for all values of θ where cos θ ≠ 0.
In the following Explore, you will derive another identity based on the Pythagorean Theorem, which is
why the identity is known as a Pythagorean identity.
A) The terminal side of an angle θ intersects the unit circle at the point (a, b) as shown. Write a and b
in terms of trigonometric functions involving θ.
B) Apply the Pythagorean theorem to the right triangle in the
diagram. Note that when a trigonometric function is squared,
the exponent is typically written immediately after the name of
the function. For instance, (sin 𝜃)2 = sin2 𝜃
Pythagorean Identity
𝜋
C) Confirm the Pythagorean identity for 𝜃 = 3 .
D) Confirm the Pythagorean identity for 𝜃 =
3𝜋
4
.
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H. Algebra 2
17.3 Notes
Finding the Value of the Other Trigonometric Functions Given the Value of sin θ or cos θ
Learning Target G: I can find the value of the other trigonometric functions given the value of sin 𝜃, cos 𝜃, or
tan 𝜃.
You can rewrite the identity sin2 𝜃 + cos2 𝜃 = 1 to express one trigonometric function in terms of the
other. As shown, each alternate version of the identity involves both positive and negative square
roots. You can determine which sign to use based on knowing the quadrant in which the terminal side
of θ lies.
Example 1 Find the approximate value of each trigonometric function.
A) Given that sin 𝜃 = 0.766 𝑤ℎ𝑒𝑟𝑒 0 < 𝜃 <
𝜋
2
, find cos 𝜃 .
B) Given that cos 𝜃 = −0.906 𝑤ℎ𝑒𝑟𝑒 𝜋 < 𝜃 <
3𝜋
C) Given that sin 𝜃 = −0.644 𝑤ℎ𝑒𝑟𝑒 𝜋 < 𝜃 <
3𝜋
D) Given that cos 𝜃 = −0.994 𝑤ℎ𝑒𝑟𝑒
𝜋
2
2
2
, find sin 𝜃 .
, find cos 𝜃 .
< 𝜃 < 𝜋 , find sin 𝜃 . Then find tan 𝜃
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H. Algebra 2
17.3 Notes
Finding the Value of Other Trigonometric Functions Given the Value of 𝒕𝒂𝒏 𝜽
If you multiply both sides of the identity tan 𝜃 =
sin 𝜃
cos 𝜃
by 𝑐𝑜𝑠𝜃, you get the identity 𝑐𝑜𝑠𝜃𝑡𝑎𝑛𝜃 = 𝑠𝑖𝑛𝜃,
or 𝑠𝑖𝑛𝜃 = 𝑐𝑜𝑠𝜃 𝑡𝑎𝑛𝜃. Also, if you divide both sides of 𝑠𝑖𝑛𝜃 = 𝑐𝑜𝑠𝜃 𝑡𝑎𝑛𝜃 by 𝑡𝑎𝑛𝜃, you get the
sin 𝜃
identitycos 𝜃 = tan 𝜃. You can use the first of these identities to find the sine and cosine of an angle
when you know the tangent.
Example 2. Find the approximate value of each trigonometric function.
𝜋
A) Given that tan 𝜃 ≈ −2.327 where 2 < 𝜃 < 𝜋, find the values of 𝑠𝑖𝑛𝜃 and 𝑐𝑜𝑠𝜃.
B) Given that tan 𝜃 ≈ −4.366 where
3𝜋
2
< 𝜃 < 2𝜋, find the values of sinθ and cosθ.
C) Given that tan 𝜃 ≈ 3.454 where < 𝜃 <
3𝜋
2
, find the values of sin 𝜃 and cos 𝜃.
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