Q: If $$\sin(\theta) = \frac{\sqrt{2}}{2}$$, what are all possible solutions for $$\theta in$$ $$[0, 2\pi)$$?

$$\theta = \frac{\pi}{4} + 2\pi n,\ \frac{3\pi}{4} + 2\pi n$$, where $$n is an $$integer

Question 1

Evaluate $$	an$$^{-1}$$(	an \frac{3\pi}{4})$$ and select the correct simplified value within the principal range.

Accepted Answer

$$-\frac{\pi}{4}$$

Question 2

What is the value of $$\cos$$^{-1}$$(-1)$$?

Accepted Answer

$$\pi$$

Question 3

If $$\sin(	heta) = \frac{\sqrt{2}}{2}$$, what are all possible solutions for $$	heta in$$ $$[0, 2\pi)$$?

Accepted Answer

$$	heta = \frac{\pi}{4} + 2\pi n,\ \frac{3\pi}{4} + 2\pi n$$, where $$n is an $$integer

Question 4

Is the equation $$\cot x \cos x = \cot x + \frac{2}{\sin x} - 1 an $$identity?

Accepted Answer

No

Question 5

Using sum and difference identities, what is the exact value of $$\sin(75^\circ)$$?

Accepted Answer

$$\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}$$

Question 6

Evaluate $$\cos(2	heta) if$$ $$	heta = \frac{\pi}{4}$$ using double angle identities.

Accepted Answer

$$0$$

Question 7

Which identity expresses $$\sin^2$(	heta) in $$terms of $$\cos(	heta)$$?

Accepted Answer

$$1 - \cos^2$(	heta)$$

Question 8

Which statement best describes the Law of Sines?

Accepted Answer

The ratio of the sine of an angle to its opposite side is constant for all angles in a triangle.

Question 9

Which formula represents the Law of Cosines for finding side $$c in a $$triangle?

Accepted Answer

$$c^2$$ = $$a^2$$ + $$b^2 - 2ab$$ \cos(C)$$

Question 10

Which formula gives the area of a triangle with sides $$b$$ and $$c$$ and included angle $$A$$?

Accepted Answer

$$\frac{1}{2}bc\sin(A)$$

Question 11

A vector makes a $$45^\circ$$ angle with the east direction. What is the direction of the vector?

Accepted Answer

Northeast

Question 12

What is the component form of a vector with initial point $$(0, 0)$$ and terminal point $$(5, 5)$$?

Accepted Answer

$$(5, 5)$$

Question 13

A vector has components $$(3, -4). In $$which quadrant is the vector located?

Accepted Answer

Fourth quadrant

Question 14

Is the vector $$0.5\mathbf{i} + 0.5\mathbf{j} a $$unit vector?

Accepted Answer

No, because its magnitude is not 1

Question 15

Given $$\mathbf{a} = 9\mathbf{i} - 4\mathbf{j}$$, $$\mathbf{b} = \mathbf{i} + 5\mathbf{j}$$, $$\mathbf{c} = 8\mathbf{i} + 3\mathbf{j}$$, compute $$\mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}.$$

Accepted Answer

$$104$$

Question 16

Calculate the cross product of $$\mathbf{u} = (1, 2, 3)$$ and $$\mathbf{v} = (4, 5, 6)$$ using the matrix method.

Accepted Answer

$$(-6, 3, -6)$$

Question 17

In the polar coordinate system, what is the 'pole'?

Accepted Answer

The origin point, equivalent to $$(0,0) in $$Cartesian coordinates.

Question 18

What are the rectangular coordinates of the polar point $$(0, 0)$$?

Accepted Answer

$$(0, 0)$$

Question 19

What is the graph shape of the equation $$r = 6 \cos(	heta)$$ after converting to rectangular form?

Accepted Answer

Circle

Question 20

How many petals does the rose curve $$r = 3 \sin 4	heta$$ have?

Accepted Answer

8 petals

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