Question

Please see the picture below. I'll only need b c and d

Answer 1

Given:

• cotθ = -3

,

• secθ < 0

,

• 0 ≤ θ < 2π

Here the cot value of the angle is negative.

The cotangent function is negative in quadrants II and IV.

Also, secθ < 0, which means it is negative.

Secant function is negative in II and III quadrants.

Therefore, the angle will be in quadrant II.

Let's find the exact values of the following:

• (a). sin(2θ)

Apply the double angle formula:

`sin(2\theta)=2sin\theta cos\theta=(2tan\theta)/(1+tan^2\theta)`

Where:

`tan\theta=(1)/(cot\theta)=-(1)/(3)`

Thus, we have:

`\begin{gathered} sin(2\theta)=(2*(-(1)/(3)))/(1+(-(1)/(3))^2) \\ \\ sin(2\theta)=(-(2)/(3))/(1+(1)/(9))=(-(2)/(3))/((9+1)/(9))=(-(2)/(3))/((10)/(9)) \\ \\ sin(2\theta)=-(2)/(3)*(9)/(10) \\ \\ sin(2\theta)=-(3)/(5) \\ \\ \text{ Sine is positive in quadrant II:} \\ sin(2\theta)=(3)/(5) \end{gathered}`

• cos(2θ):

Apply the formula:

`cos(2\theta)=(1-tan^2\theta)/(1+tan^2\theta)`

Thus, we have:

`\begin{gathered} cos(2\theta)=(1-(-(1)/(3))^2)/(1+(-(1)/(3))^2) \\ \\ cos(2\theta)=(1-(1)/(9))/(1+(1)/(9)) \\ \\ cos(2\theta)=((9-1)/(9))/((9+1)/(9))=((8)/(9))/((10)/(9))=(8)/(9)*(9)/(10)=(4)/(5) \\ \\ cos(2\theta)=(4)/(5) \\ \text{ } \\ \text{ Cosine is negative in quadrant II>} \\ cosine(2\theta)=-(4)/(5) \end{gathered}`

• (c). sin(θ/2):

Apply the formula:

`cos\theta=1-2sin^2((\theta)/(2))`

Where:

`tan\theta=(opposite)/(adjacent)=-(1)/(3)`

Now, let's find the hypotenuse using Pythagorean Theorem:

`√(1^2+3^2)=√(1+9)=√(10)`

Thus, we have:

`cos\theta=(adjacent)/(hypotenuse)=-(3)/(√(10))`

Now, the function will be:

`\begin{gathered} cos\theta=1-2sin^2((\theta)/(2)) \\ \\ -(3)/(√(10))=1-2sin^2((\theta)/(2)) \\ \\ 2sin^2((\theta)/(2))=1+(3)/(√(10)) \\ \\ 2sin^2((\theta)/(2))=(10+3√(10))/(10) \\ \\ sin^2((\theta)/(2))=(10+3√(10))/(20) \\ \\ sin((\theta)/(2))=\sqrt{(10+3√(10))/(20)} \end{gathered}`

• (d). cos(,(θ/2)):

`\begin{gathered} 2cos\theta=2cos^2((\theta)/(2))-1 \\ \\ cos(\theta)/(2)=\sqrt{(1+cos\theta)/(2)}=\sqrt{(1-(3)/(√(10)))/(2)} \end{gathered}`

ANSWER:

`\begin{gathered} (a).\text{ }(3)/(5) \\ \\ \\ (b).\text{ -}(4)/(5) \\ \\ \\ (c).\text{ }\sqrt{(10+3√(10))/(20)} \\ \\ \\ (d).\text{ }\sqrt{(1-(3)/(√(10)))/(2)} \end{gathered}`