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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 π¦ where the terminal ray of an angle ΞΈ intersects the circle are x = cos ΞΈ and y = sin ΞΈ, and that tan ΞΈ = π₯ . sin π Combining these facts gives the identity tan π = cos π, which is true for all values of ΞΈ where cos ΞΈ β 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 ΞΈ intersects the unit circle at the point (a, b) as shown. Write a and b in terms of trigonometric functions involving ΞΈ. 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 π)2 = sin2 π Pythagorean Identity π C) Confirm the Pythagorean identity for π = 3 . D) Confirm the Pythagorean identity for π = 3π 4 . 1 H. Algebra 2 17.3 Notes Finding the Value of the Other Trigonometric Functions Given the Value of sin ΞΈ or cos ΞΈ Learning Target G: I can find the value of the other trigonometric functions given the value of sin π, cos π, or tan π. You can rewrite the identity sin2 π + cos2 π = 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 ΞΈ lies. Example 1 Find the approximate value of each trigonometric function. A) Given that sin π = 0.766 π€βπππ 0 < π < π 2 , find cos π . B) Given that cos π = β0.906 π€βπππ π < π < 3π C) Given that sin π = β0.644 π€βπππ π < π < 3π D) Given that cos π = β0.994 π€βπππ π 2 2 2 , find sin π . , find cos π . < π < π , find sin π . Then find tan π 2 H. Algebra 2 17.3 Notes Finding the Value of Other Trigonometric Functions Given the Value of πππ π½ If you multiply both sides of the identity tan π = sin π cos π by πππ π, you get the identity πππ ππ‘πππ = π πππ, or π πππ = πππ π π‘πππ. Also, if you divide both sides of π πππ = πππ π π‘πππ by π‘πππ, you get the sin π identitycos π = tan π. 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 π β β2.327 where 2 < π < π, find the values of π πππ and πππ π. B) Given that tan π β β4.366 where 3π 2 < π < 2π, find the values of sinΞΈ and cosΞΈ. C) Given that tan π β 3.454 where < π < 3π 2 , find the values of sin π and cos π. 3