Question

writing a trigonometric identity
go to bigideasmath.com for an interactive tool to investigate this exploration.
work with a partner. in the figure, the point (x, y) is on a
circle of radius c with center at the origin.
a. write an equation that relates a, b, and c.
b. write expressions for the sine and cosine ratios of angle θ.
c. use the results from parts (a) and (b) to find the sum of sin²θ and cos²θ. what do you observe?

Explanation:

Step1: Recall trigonometric ratio definitions

For a right - triangle, \(\sin\theta=\frac{b}{c}\) and \(\cos\theta=\frac{a}{c}\).

Step2: Calculate \(\sin^{2}\theta+\cos^{2}\theta\)

Substitute the expressions from Step1: \(\sin^{2}\theta+\cos^{2}\theta = (\frac{b}{c})^{2}+(\frac{a}{c})^{2}=\frac{b^{2}+a^{2}}{c^{2}}\).

Step3: Use the result from part (a)

Since \(a^{2}+b^{2}=c^{2}\) (from part (a)), then \(\frac{b^{2}+a^{2}}{c^{2}}=\frac{c^{2}}{c^{2}} = 1\).

Answer:

\(\sin^{2}\theta+\cos^{2}\theta=1\)