Chapter 7.4, Problem 107AYU

Write a few paragraphs outlining your strategy for establishing identities.

To determine

To find: The strategy for establishing an identity.

Answer to Problem 107AYU

Solution:

The strategy for establishing an identity is,

Consider a circle with radius $R$.

Let $P\left(x\text{, }y\right)$ be a terminal point on the circle which makes an angle of $\text{θ}$ with the Centre of the circle (origin).

By definition of six trigonometric functions.

We can obtain the identities.

For example Pythagorean identity.

${\text{sin}}^2\theta +{\text{ cos}}^2\theta =\frac{y^2}{r^2}+\frac{x^2}{r^2}$

Adding up the quotients.

${\text{sin}}^2\theta +{\text{ cos}}^2\theta =\frac{y^2+x^2}{r^2}$

By the equation of the circle.

$x^2+y^2=r^2$

${\text{sin}}^2\theta +{\text{ cos}}^2\theta =\frac{r^2}{r^2}=1$

Explanation of Solution

The strategy for establishing an identity is:

Consider a circle with radius $R$.

Let $P\left(x\text{, }y\right)$ be a terminal point on the circle which makes an angle of $\text{θ}$ with the Centre of the circle (origin).

By definition of six trigonometric functions.

We can obtain the identities.

For example Pythagorean identity.

${\text{sin}}^2\theta +{\text{ cos}}^2\theta =\frac{y^2}{r^2}+\frac{x^2}{r^2}$

Adding up the quotients.

${\text{sin}}^2\theta +{\text{ cos}}^2\theta =\frac{y^2+x^2}{r^2}$

By the equation of the circle.

$x^2+y^2=r^2$

${\text{sin}}^2\theta +{\text{ cos}}^2\theta =\frac{r^2}{r^2}=1$