Question

Solve without using a calculator. Letter f please.

Answer 1

Answer:

B: $-\frac{\sqrt{2} }{2}$

F: $\frac{pi}{3}$, or 60 degrees.

Step-by-step explanation:

Note: theta = θ

A calculator isn't needed for this at all. All you need is the Unit circle, and an understanding of the difference between trig functions and inverse trig functions.

Refer to the attachment for the Unit Circle. I COULD go into how each of those values came to be, but for now, just know they were all derived from 90 degree, 45 degree, and 30 degree triangles.

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B: To answer B, you know you're using the Trig function sine, which is of the form sinθ = x. Note that you KNOW the angle, and you're looking for x.

When in terms of a unit circle (and specifically for sine), "x" no longer represents opposite/hypotenuse, but now represents "y" in the format of (x, y) aka (cosθ, sinθ). They are the format of the values you see in the Unit circle! Now, search the circle for 5pi/4. . . you should find it lands on the angle 225 degrees, which is of the form ($-\frac{\sqrt{2} }{2}$, $-\frac{\sqrt{2} }{2}$ ) . In the format of (x , y) or (cosθ, sinθ), we know we're looking for the "y" value, so $-\frac{\sqrt{2} }{2}$.

F: To answer F, know you're using the INVERSE trig function sin, which is of the form $theta = arcsin x$ or $theta = sin^{-1} x$. Note that you KNOW the "x" and you're looking for the angle.

When in terms of the unit circle and inverse trig, the "x" becomes the corresponding value to the VALUE at that angle. Since we are in terms of sine, we know the corresponding value is the "y" value for the angle, so, look at the unit circle and find ALL the angles where you see the "y" value of $\frac{\sqrt{3} }{2}$.

This occurs in two places: 120 degrees, and 60 degrees.

Restrictions:
BUT DON'T GET CARELESS! Something you must ALWAYS remember are the restrictions on sine, cosine, and tangent, and how these restrictions translate to their inverse counterparts. For arcsin, you are restricted to answers ONLY in quadrant I and IV, so on the right side of the y-axis. This means the answer 120 degrees becomes invalid, because it is in quadrant II.

Therefore, the only remaining valid answer is pi/3 or 60 degrees.