Algebra 2 Common Core

Published by Prentice Hall
ISBN 10: 0133186024
ISBN 13: 978-0-13318-602-4

Chapter 14 - Trigonometric Identities and Equations - 14-1 Trigonometric Identities - Lesson Check: 5

Answer

In a Right Triangle, where x is height, y is base and z is hypotenuse, Pythagoras theorem is: $x^{2}+y^{2}=z^{2}$ Refer to the step-by-step part for the explanation.

Work Step by Step

Let the angle between z(hypotenuse) and y(base) in a right angled triangle be $\theta$ Step 1: By identity, $\sin^{2}\theta+\cos^{2}\theta=1$ Step 2: By definition, $\sin\theta= \frac{\text{OppositeSide}}{\text{Hypotenuse}}$ and $\cos\theta= \frac{\text{AdjacentSide}}{\text{Hypotenuse}}$ Therefore, $\sin^{2}\theta+\cos^{2}\theta= \frac{\text{OppositeSide}^{2}}{\text{Hypotenuse}^{2}}+\frac{\text{AdjacentSide}^{2}}{\text{Hypotenuse}^{2}}$ =1 Step 3: The opposite side is $x$ and adjacent side is $y$, therefore, $\sin^{2}\theta+\cos^{2}\theta= \frac{x^{2}}{z^{2}}+\frac{y^{2}}{z^{2}}=1 $ Step 4: Add the numerators and copy the common denominator: $\frac{x^{2}+y^{2}}{z^{2}}=1$ Step 5: Cross multiplication, $x^{2}+y^{2}=z^{2}$ (Pythagorean Theorem)
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