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solvefor θ,r^2cos^2(θ)+r^2sin^2(θ)<= 1

Solution

solve for

θ, r^{2} cos^{2} (θ) + r^{2} sin^{2} (θ) \leq 1

Solution

- 1 \leq θ \leq 1

Solution steps

r^{2} cos^{2} (θ) + r^{2} sin^{2} (θ) \leq 1

Use the following identity:

cos^{2} (x) + sin^{2} (x) = 1

Therefore

cos^{2} (x) = 1 - sin^{2} (x)

r^{2} (1 - sin^{2} (θ)) + r^{2} sin^{2} (θ) \leq 1

Simplify

r^{2} (1 - sin^{2} (θ)) + r^{2} sin^{2} (θ) : r^{2}

r^{2} (1 - sin^{2} (θ)) + r^{2} sin^{2} (θ)

Expand

r^{2} (1 - sin^{2} (θ)) : r^{2} - r^{2} sin^{2} (θ)

r^{2} (1 - sin^{2} (θ))

Apply the distributive law:

a (b - c) = ab - a c

a = r^{2}, b = 1, c = sin^{2} (θ)

= r^{2} \cdot 1 - r^{2} sin^{2} (θ)

= 1 \cdot r^{2} - r^{2} sin^{2} (θ)

Multiply:

1 \cdot r^{2} = r^{2}

= r^{2} - r^{2} sin^{2} (θ)

= r^{2} - r^{2} sin^{2} (θ) + r^{2} sin^{2} (θ)

Add similar elements:

- r^{2} sin^{2} (θ) + r^{2} sin^{2} (θ) = 0

= r^{2}

r^{2} \leq 1

Rewrite in standard form

r^{2} \leq 1

Subtract

1

from both sides

r^{2} - 1 \leq 1 - 1

Simplify

r^{2} - 1 \leq 0

r^{2} - 1 \leq 0

Factor

r^{2} - 1 : (r + 1) (r - 1)

r^{2} - 1

Rewrite

1

1^{2}

= r^{2} - 1^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

r^{2} - 1^{2} = (r + 1) (r - 1)

= (r + 1) (r - 1)

(r + 1) (r - 1) \leq 0

Identify the intervals

Find the signs of the factors of

(r + 1) (r - 1)

Find the signs of

r + 1

r + 1 = 0 : r = - 1

r + 1 = 0

Move

1

to the right side

r + 1 = 0

Subtract

1

from both sides

r + 1 - 1 = 0 - 1

Simplify

r = - 1

r = - 1

r + 1 < 0 : r < - 1

r + 1 < 0

Move

1

to the right side

r + 1 < 0

Subtract

1

from both sides

r + 1 - 1 < 0 - 1

Simplify

r < - 1

r < - 1

r + 1 > 0 : r > - 1

r + 1 > 0

Move

1

to the right side

r + 1 > 0

Subtract

1

from both sides

r + 1 - 1 > 0 - 1

Simplify

r > - 1

r > - 1

Find the signs of

r - 1

r - 1 = 0 : r = 1

r - 1 = 0

Move

1

to the right side

r - 1 = 0

Add

1

to both sides

r - 1 + 1 = 0 + 1

Simplify

r = 1

r = 1

r - 1 < 0 : r < 1

r - 1 < 0

Move

1

to the right side

r - 1 < 0

Add

1

to both sides

r - 1 + 1 < 0 + 1

Simplify

r < 1

r < 1

r - 1 > 0 : r > 1

r - 1 > 0

Move

1

to the right side

r - 1 > 0

Add

1

to both sides

r - 1 + 1 > 0 + 1

Simplify

r > 1

r > 1

Summarize in a table:

r + 1 r - 1 (r + 1) (r - 1) r < - 1 - - + r = - 1 0 - 0 - 1 < r < 1 + - - r = 1 + 00 r > 1 + + +

Identify the intervals that satisfy the required condition:

\leq 0

r = - 1 or - 1 < r < 1 or r = 1

Merge Overlapping Intervals

- 1 \leq r < 1 or r = 1

The union of two intervals is the set of numbers which are in either interval

r = - 1

- 1 < r < 1

- 1 \leq r < 1

The union of two intervals is the set of numbers which are in either interval

- 1 \leq r < 1

r = 1

- 1 \leq r \leq 1

- 1 \leq r \leq 1

- 1 \leq θ \leq 1

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