There are many different forms the equation of a circle can take, but they usually reduce to some manipulation of the following:
{eq}\begin{align*} (x-a)^2 + (y-b)^2 &= r^2 \end{align*} {/eq}
where {eq}r {/eq} is the radius of the circle and {eq}(a,b) {/eq} is the center. When the center is the origin, we have {eq}a=b=0 {/eq} and so
{eq}\begin{align*} x^2 + y^2 &= r^2 \end{align*} {/eq}
Then since {eq}r^2 \cos^2 t + r^2 \sin^2 t = r^2 {/eq}, we can write
{eq}\begin{align*} x &= r \cos t \\ y &= r \sin t \end{align*} {/eq}
where {eq}t \in [0,2\pi] {/eq} gets us once around the circle.
We can flip these trig functions around or solve the sum of two squares explicitly for {eq}x {/eq} or {eq}y {/eq} and we haven't changed a thing at all.
Answer and Explanation:1
Part A
This is the classic sine-cosine circle, but with the trig function flipped. All that does is change where we start (at the top instead of the...
Step 7.1
Write the expression using exponents.
Step 7.1.1
Draw a triangle in the plane with vertices , , and the origin. Then is the angle between the positive x-axis and the ray beginning at the origin and passing through . Therefore, is .
Step 7.1.2
Step 7.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 7.3
Step 7.3.1
Write as a fraction with a common denominator.
Step 7.3.2
Combine the numerators over the common denominator.
Step 7.3.3
Write as a fraction with a common denominator.
Step 7.3.4
Combine the numerators over the common denominator.
Step 7.3.5
Step 7.3.6
Step 7.4
Step 7.4.1
Factor the perfect power out of .
Step 7.4.2
Factor the perfect power out of .
Step 7.4.3
Step 7.5
Pull terms out from under the radical.
Step 7.6
Step 7.7
Cancel the common factor of .
Step 7.7.1
Cancel the common factor.
Step 7.7.2
Step 7.1
Write the expression using exponents.
Step 7.1.1
Draw a triangle in the plane with vertices , , and the origin. Then is the angle between the positive x-axis and the ray beginning at the origin and passing through . Therefore, is .
Step 7.1.2
Step 7.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 7.3
Step 7.3.1
Write as a fraction with a common denominator.
Step 7.3.2
Combine the numerators over the common denominator.
Step 7.3.3
Write as a fraction with a common denominator.
Step 7.3.4
Combine the numerators over the common denominator.
Step 7.3.5
Step 7.3.6
Step 7.4
Step 7.4.1
Factor the perfect power out of .
Step 7.4.2
Factor the perfect power out of .
Step 7.4.3
Step 7.5
Pull terms out from under the radical.
Step 7.6
Step 7.7
Cancel the common factor of .
Step 7.7.1
Cancel the common factor.
Step 7.7.2