Trigonometry Examples

$y = 4 \cos (\frac{3 θ}{2})$

Step 1

Find the first derivative of the function.

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Step 1.1

Since $4$ is constant with respect to $θ$ , the derivative of $4 \cos (\frac{3 θ}{2})$ with respect to $θ$ is $4 \frac{d}{d θ} [\cos (\frac{3 θ}{2})]$ .

$4 \frac{d}{d θ} [\cos (\frac{3 θ}{2})]$

Step 1.2

Differentiate using the chain rule, which states that $\frac{d}{d θ} [f (g (θ))]$ is $f' (g (θ)) g' (θ)$ where $f (θ) = \cos (θ)$ and $g (θ) = \frac{3 θ}{2}$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $\frac{3 θ}{2}$ .

$4 (\frac{d}{d u} [\cos (u)] \frac{d}{d θ} [\frac{3 θ}{2}])$

Step 1.2.2

The derivative of $\cos (u)$ with respect to $u$ is $- \sin (u)$ .

$4 (- \sin (u) \frac{d}{d θ} [\frac{3 θ}{2}])$

Step 1.2.3

Replace all occurrences of $u$ with $\frac{3 θ}{2}$ .

$4 (- \sin (\frac{3 θ}{2}) \frac{d}{d θ} [\frac{3 θ}{2}])$

Step 1.3

Differentiate.

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Step 1.3.1

Multiply $- 1$ by $4$ .

$- 4 (\sin (\frac{3 θ}{2}) \frac{d}{d θ} [\frac{3 θ}{2}])$

Step 1.3.2

Since $\frac{3}{2}$ is constant with respect to $θ$ , the derivative of $\frac{3 θ}{2}$ with respect to $θ$ is $\frac{3}{2} \frac{d}{d θ} [θ]$ .

$- 4 \sin (\frac{3 θ}{2}) (\frac{3}{2} \frac{d}{d θ} [θ])$

Step 1.3.3

Simplify terms.

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Step 1.3.3.1

Combine $\frac{3}{2}$ and $- 4$ .

$\frac{3 \cdot - 4}{2} \sin (\frac{3 θ}{2}) \frac{d}{d θ} [θ]$

Step 1.3.3.2

Multiply $3$ by $- 4$ .

$\frac{- 12}{2} \sin (\frac{3 θ}{2}) \frac{d}{d θ} [θ]$

Step 1.3.3.3

Combine $\frac{- 12}{2}$ and $\sin (\frac{3 θ}{2})$ .

$\frac{- 12 \sin (\frac{3 θ}{2})}{2} \frac{d}{d θ} [θ]$

Step 1.3.3.4

Cancel the common factor of $- 12$ and $2$ .

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Step 1.3.3.4.1

Factor $2$ out of $- 12 \sin (\frac{3 θ}{2})$ .

$\frac{2 (- 6 \sin (\frac{3 θ}{2}))}{2} \frac{d}{d θ} [θ]$

Step 1.3.3.4.2

Cancel the common factors.

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Step 1.3.3.4.2.1

Factor $2$ out of $2$ .

$\frac{2 (- 6 \sin (\frac{3 θ}{2}))}{2 (1)} \frac{d}{d θ} [θ]$

Step 1.3.3.4.2.2

Cancel the common factor.

$\frac{2 (- 6 \sin (\frac{3 θ}{2}))}{2 \cdot 1} \frac{d}{d θ} [θ]$

Step 1.3.3.4.2.3

Rewrite the expression.

$\frac{- 6 \sin (\frac{3 θ}{2})}{1} \frac{d}{d θ} [θ]$

Step 1.3.3.4.2.4

Divide $- 6 \sin (\frac{3 θ}{2})$ by $1$ .

$- 6 \sin (\frac{3 θ}{2}) \frac{d}{d θ} [θ]$

Step 1.3.4

Differentiate using the Power Rule which states that $\frac{d}{d θ} [θ^{n}]$ is $n θ^{n - 1}$ where $n = 1$ .

$- 6 \sin (\frac{3 θ}{2}) \cdot 1$

Step 1.3.5

Multiply $- 6$ by $1$ .

$- 6 \sin (\frac{3 θ}{2})$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $- 6$ is constant with respect to $θ$ , the derivative of $- 6 \sin (\frac{3 θ}{2})$ with respect to $θ$ is $- 6 \frac{d}{d θ} [\sin (\frac{3 θ}{2})]$ .

$f'' (θ) = - 6 \frac{d}{d θ} \sin (\frac{3 θ}{2})$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d θ} [f (g (θ))]$ is $f' (g (θ)) g' (θ)$ where $f (θ) = \sin (θ)$ and $g (θ) = \frac{3 θ}{2}$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $\frac{3 θ}{2}$ .

$f'' (θ) = - 6 (\frac{d}{d u} (\sin (u)) \frac{d}{d θ} (\frac{3 θ}{2}))$

Step 2.2.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

$f'' (θ) = - 6 (\cos (u) \frac{d}{d θ} (\frac{3 θ}{2}))$

Step 2.2.3

Replace all occurrences of $u$ with $\frac{3 θ}{2}$ .

$f'' (θ) = - 6 (\cos (\frac{3 θ}{2}) \frac{d}{d θ} (\frac{3 θ}{2}))$

Step 2.3

Differentiate.

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Step 2.3.1

Since $\frac{3}{2}$ is constant with respect to $θ$ , the derivative of $\frac{3 θ}{2}$ with respect to $θ$ is $\frac{3}{2} \frac{d}{d θ} [θ]$ .

$f'' (θ) = - 6 \cos (\frac{3 θ}{2}) (\frac{3}{2} \cdot \frac{d}{d θ} (θ))$

Step 2.3.2

Simplify terms.

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Step 2.3.2.1

Combine $\frac{3}{2}$ and $- 6$ .

$f'' (θ) = \frac{3 \cdot - 6}{2} \cdot \cos (\frac{3 θ}{2}) \frac{d}{d θ} (θ)$

Step 2.3.2.2

Multiply $3$ by $- 6$ .

$f'' (θ) = \frac{- 18}{2} \cdot \cos (\frac{3 θ}{2}) \frac{d}{d θ} (θ)$

Step 2.3.2.3

Combine $\frac{- 18}{2}$ and $\cos (\frac{3 θ}{2})$ .

$f'' (θ) = \frac{- 18 \cos (\frac{3 θ}{2})}{2} \cdot \frac{d}{d θ} (θ)$

Step 2.3.2.4

Cancel the common factor of $- 18$ and $2$ .

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Step 2.3.2.4.1

Factor $2$ out of $- 18 \cos (\frac{3 θ}{2})$ .

$f'' (θ) = \frac{2 (- 9 \cos (\frac{3 θ}{2}))}{2} \cdot \frac{d}{d θ} (θ)$

Step 2.3.2.4.2

Cancel the common factors.

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Step 2.3.2.4.2.1

Factor $2$ out of $2$ .

$f'' (θ) = \frac{2 (- 9 \cos (\frac{3 θ}{2}))}{2 (1)} \cdot \frac{d}{d θ} (θ)$

Step 2.3.2.4.2.2

Cancel the common factor.

$f'' (θ) = \frac{2 (- 9 \cos (\frac{3 θ}{2}))}{2 \cdot 1} \cdot \frac{d}{d θ} (θ)$

Step 2.3.2.4.2.3

Rewrite the expression.

$f'' (θ) = \frac{- 9 \cos (\frac{3 θ}{2})}{1} \cdot \frac{d}{d θ} (θ)$

Step 2.3.2.4.2.4

Divide $- 9 \cos (\frac{3 θ}{2})$ by $1$ .

$f'' (θ) = - 9 \cos (\frac{3 θ}{2}) \frac{d}{d θ} θ$

Step 2.3.3

Differentiate using the Power Rule which states that $\frac{d}{d θ} [θ^{n}]$ is $n θ^{n - 1}$ where $n = 1$ .

$f'' (θ) = - 9 \cos (\frac{3 θ}{2}) \cdot 1$

Step 2.3.4

Multiply $- 9$ by $1$ .

$f'' (θ) = - 9 \cos (\frac{3 θ}{2})$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$- 6 \sin (\frac{3 θ}{2}) = 0$

Step 4

Divide each term in $- 6 \sin (\frac{3 θ}{2}) = 0$ by $- 6$ and simplify.

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Step 4.1

Divide each term in $- 6 \sin (\frac{3 θ}{2}) = 0$ by $- 6$ .

$\frac{- 6 \sin (\frac{3 θ}{2})}{- 6} = \frac{0}{- 6}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 6$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 6 \sin (\frac{3 θ}{2})}{- 6} = \frac{0}{- 6}$

Step 4.2.1.2

Divide $\sin (\frac{3 θ}{2})$ by $1$ .

$\sin (\frac{3 θ}{2}) = \frac{0}{- 6}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $- 6$ .

$\sin (\frac{3 θ}{2}) = 0$

Step 5

Take the inverse sine of both sides of the equation to extract $θ$ from inside the sine.

$\frac{3 θ}{2} = \arcsin (0)$

Step 6

Simplify the right side.

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Step 6.1

The exact value of $\arcsin (0)$ is $0$ .

$\frac{3 θ}{2} = 0$

Step 7

Set the numerator equal to zero.

$3 θ = 0$

Step 8

Divide each term in $3 θ = 0$ by $3$ and simplify.

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Step 8.1

Divide each term in $3 θ = 0$ by $3$ .

$\frac{3 θ}{3} = \frac{0}{3}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $3$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{3 θ}{3} = \frac{0}{3}$

Step 8.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{0}{3}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Divide $0$ by $3$ .

$θ = 0$

Step 9

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$\frac{3 θ}{2} = π - 0$

Step 10

Solve for $θ$ .

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Step 10.1

Multiply both sides of the equation by $\frac{2}{3}$ .

$\frac{2}{3} \cdot \frac{3 θ}{2} = \frac{2}{3} (π - 0)$

Step 10.2

Simplify both sides of the equation.

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Step 10.2.1

Simplify the left side.

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Step 10.2.1.1

Simplify $\frac{2}{3} \cdot \frac{3 θ}{2}$ .

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Step 10.2.1.1.1

Cancel the common factor of $2$ .

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Step 10.2.1.1.1.1

Cancel the common factor.

$\frac{2}{3} \cdot \frac{3 θ}{2} = \frac{2}{3} (π - 0)$

Step 10.2.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 θ) = \frac{2}{3} (π - 0)$

Step 10.2.1.1.2

Cancel the common factor of $3$ .

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Step 10.2.1.1.2.1

Factor $3$ out of $3 θ$ .

$\frac{1}{3} (3 (θ)) = \frac{2}{3} (π - 0)$

Step 10.2.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 θ) = \frac{2}{3} (π - 0)$

Step 10.2.1.1.2.3

Rewrite the expression.

$θ = \frac{2}{3} (π - 0)$

Step 10.2.2

Simplify the right side.

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Step 10.2.2.1

Simplify $\frac{2}{3} (π - 0)$ .

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Step 10.2.2.1.1

Subtract $0$ from $π$ .

$θ = \frac{2}{3} π$

Step 10.2.2.1.2

Combine $\frac{2}{3}$ and $π$ .

$θ = \frac{2 π}{3}$

Step 11

The solution to the equation $\frac{3 θ}{2} = 0$ .

$θ = 0, \frac{2 π}{3}$

Step 12

Evaluate the second derivative at $θ = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 9 \cos (\frac{3 (0)}{2})$

Step 13

Evaluate the second derivative.

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Step 13.1

Cancel the common factor of $0$ and $2$ .

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Step 13.1.1

Factor $2$ out of $3 (0)$ .

$- 9 \cos (\frac{2 (3 \cdot (0))}{2})$

Step 13.1.2

Cancel the common factors.

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Step 13.1.2.1

Factor $2$ out of $2$ .

$- 9 \cos (\frac{2 (3 \cdot (0))}{2 (1)})$

Step 13.1.2.2

Cancel the common factor.

$- 9 \cos (\frac{2 (3 \cdot (0))}{2 \cdot 1})$

Step 13.1.2.3

Rewrite the expression.

$- 9 \cos (\frac{3 \cdot (0)}{1})$

Step 13.1.2.4

Divide $3 \cdot (0)$ by $1$ .

$- 9 \cos (3 \cdot (0))$

Step 13.2

Multiply $3$ by $0$ .

$- 9 \cos (0)$

Step 13.3

The exact value of $\cos (0)$ is $1$ .

$- 9 \cdot 1$

Step 13.4

Multiply $- 9$ by $1$ .

$- 9$

Step 14

$θ = 0$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$θ = 0$ is a local maximum

Step 15

Find the y-value when $θ = 0$ .

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Step 15.1

Replace the variable $θ$ with $0$ in the expression.

$f (0) = 4 \cos (\frac{3 (0)}{2})$

Step 15.2

Simplify the result.

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Step 15.2.1

Cancel the common factor of $0$ and $2$ .

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Step 15.2.1.1

Factor $2$ out of $3 (0)$ .

$f (0) = 4 \cos (\frac{2 (3 \cdot (0))}{2})$

Step 15.2.1.2

Cancel the common factors.

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Step 15.2.1.2.1

Factor $2$ out of $2$ .

$f (0) = 4 \cos (\frac{2 (3 \cdot (0))}{2 (1)})$

Step 15.2.1.2.2

Cancel the common factor.

$f (0) = 4 \cos (\frac{2 (3 \cdot (0))}{2 \cdot 1})$

Step 15.2.1.2.3

Rewrite the expression.

$f (0) = 4 \cos (\frac{3 \cdot (0)}{1})$

Step 15.2.1.2.4

Divide $3 \cdot (0)$ by $1$ .

$f (0) = 4 \cos (3 \cdot (0))$

Step 15.2.2

Multiply $3$ by $0$ .

$f (0) = 4 \cos (0)$

Step 15.2.3

The exact value of $\cos (0)$ is $1$ .

$f (0) = 4 \cdot 1$

Step 15.2.4

Multiply $4$ by $1$ .

$f (0) = 4$

Step 15.2.5

The final answer is $4$ .

$y = 4$

Step 16

Evaluate the second derivative at $θ = \frac{2 π}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 9 \cos (\frac{3 (\frac{2 π}{3})}{2})$

Step 17

Evaluate the second derivative.

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Step 17.1

Combine $3$ and $\frac{2 π}{3}$ .

$- 9 \cos (\frac{\frac{3 (2 π)}{3}}{2})$

Step 17.2

Multiply $3$ by $2$ .

$- 9 \cos (\frac{\frac{6 π}{3}}{2})$

Step 17.3

Reduce the expression by cancelling the common factors.

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Step 17.3.1

Reduce the expression $\frac{6 π}{3}$ by cancelling the common factors.

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Step 17.3.1.1

Factor $3$ out of $6 π$ .

$- 9 \cos (\frac{\frac{3 (2 π)}{3}}{2})$

Step 17.3.1.2

Factor $3$ out of $3$ .

$- 9 \cos (\frac{\frac{3 (2 π)}{3 (1)}}{2})$

Step 17.3.1.3

Cancel the common factor.

$- 9 \cos (\frac{\frac{3 (2 π)}{3 \cdot 1}}{2})$

Step 17.3.1.4

Rewrite the expression.

$- 9 \cos (\frac{\frac{2 π}{1}}{2})$

Step 17.3.2

Divide $2 π$ by $1$ .

$- 9 \cos (\frac{2 π}{2})$

Step 17.4

Cancel the common factor of $2$ .

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Step 17.4.1

Cancel the common factor.

$- 9 \cos (\frac{2 π}{2})$

Step 17.4.2

Divide $π$ by $1$ .

$- 9 \cos (π)$

Step 17.5

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$- 9 (- \cos (0))$

Step 17.6

The exact value of $\cos (0)$ is $1$ .

$- 9 (- 1 \cdot 1)$

Step 17.7

Multiply $- 9 (- 1 \cdot 1)$ .

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Step 17.7.1

Multiply $- 1$ by $1$ .

$- 9 \cdot - 1$

Step 17.7.2

Multiply $- 9$ by $- 1$ .

$9$

Step 18

$θ = \frac{2 π}{3}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$θ = \frac{2 π}{3}$ is a local minimum

Step 19

Find the y-value when $θ = \frac{2 π}{3}$ .

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Step 19.1

Replace the variable $θ$ with $\frac{2 π}{3}$ in the expression.

$f (\frac{2 π}{3}) = 4 \cos (\frac{3 (\frac{2 π}{3})}{2})$

Step 19.2

Simplify the result.

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Step 19.2.1

Combine $3$ and $\frac{2 π}{3}$ .

$f (\frac{2 π}{3}) = 4 \cos (\frac{\frac{3 (2 π)}{3}}{2})$

Step 19.2.2

Multiply $3$ by $2$ .

$f (\frac{2 π}{3}) = 4 \cos (\frac{\frac{6 π}{3}}{2})$

Step 19.2.3

Reduce the expression by cancelling the common factors.

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Step 19.2.3.1

Reduce the expression $\frac{6 π}{3}$ by cancelling the common factors.

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Step 19.2.3.1.1

Factor $3$ out of $6 π$ .

$f (\frac{2 π}{3}) = 4 \cos (\frac{\frac{3 (2 π)}{3}}{2})$

Step 19.2.3.1.2

Factor $3$ out of $3$ .

$f (\frac{2 π}{3}) = 4 \cos (\frac{\frac{3 (2 π)}{3 (1)}}{2})$

Step 19.2.3.1.3

Cancel the common factor.

$f (\frac{2 π}{3}) = 4 \cos (\frac{\frac{3 (2 π)}{3 \cdot 1}}{2})$

Step 19.2.3.1.4

Rewrite the expression.

$f (\frac{2 π}{3}) = 4 \cos (\frac{\frac{2 π}{1}}{2})$

Step 19.2.3.2

Divide $2 π$ by $1$ .

$f (\frac{2 π}{3}) = 4 \cos (\frac{2 π}{2})$

Step 19.2.4

Cancel the common factor of $2$ .

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Step 19.2.4.1

Cancel the common factor.

$f (\frac{2 π}{3}) = 4 \cos (\frac{2 π}{2})$

Step 19.2.4.2

Divide $π$ by $1$ .

$f (\frac{2 π}{3}) = 4 \cos (π)$

Step 19.2.5

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$f (\frac{2 π}{3}) = 4 (- \cos (0))$

Step 19.2.6

The exact value of $\cos (0)$ is $1$ .

$f (\frac{2 π}{3}) = 4 (- 1 \cdot 1)$

Step 19.2.7

Multiply $4 (- 1 \cdot 1)$ .

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Step 19.2.7.1

Multiply $- 1$ by $1$ .

$f (\frac{2 π}{3}) = 4 \cdot - 1$

Step 19.2.7.2

Multiply $4$ by $- 1$ .

$f (\frac{2 π}{3}) = - 4$

Step 19.2.8

The final answer is $- 4$ .

$y = - 4$

Step 20

These are the local extrema for $f (θ) = 4 \cos (\frac{3 θ}{2})$ .

$(0, 4)$ is a local maxima

$(\frac{2 π}{3}, - 4)$ is a local minima

Step 21