Using Identities to Evaluate Once you've learned some trig identities, you can use them to evaluate angles exactly. Yes, you can always get a decimal approximation with your calculator, but it's good to learn the logic of using the identities, so you can exercise your skills at patternfinding and matching.
I know the exact values for the cosine of 30° and 45°, and 75° = 30° + 45°, so I'll use the anglesum identity to compute this value "exactly": You can check your answers to this sort of exercise by plugging each of the trig expression and the "exact" expression into your calculator. If the displayed values are the same (0.2588190451, in this case), then you know you've done the evaluation correctly.
Since 15 = 45 – 30, I'll use the angledifference identity for tangent: If your book prefers radians, you'll be asked to evaluate expressions like , which is the same angle as above. The sums and differences can be a bit harder to recognize in radian form: A couple of useful sums are: If you work better with degrees, then convert
radianmeasure angles to degrees, find the sums and differences, and convert
back to radians.
Since 120° is twice 60°, I'll use the doubleangle formula for sine: sin(2×60°) = 2sin(60°)cos(60°) Since 60° is a basic reference angle, I can now finish the evaluation by plugging in the values I've memorized:
Since 22.5 is half of 45, I'll use the halfangle identity for cosine: Since 22.5° is in the first quadrant, then the cosine value is positive, so I'll take the positive root for my answer:
From the sign on the sine, I could only tell that x was in QIII or QIV; that's why they had to specify the quadrant for x. By the Pythagorean Theorem, I can find the third side of the triangle from the sine's ratio value: 13^{2} = (–5)^{2}
+ (adj)^{2} So the adjacent side has a length of ±12. Since the angle x is in the third quadrant, then the "length" is –12. Plugging into the doubleangle formula, I get: In this case, I didn't really "need" the quadrant information; the squaring and subtraction took care of the signs. But sometimes you will need to pick one of two values, as in the previous example. Don't get lazy with the "plus / minus" signs. Keep track of where you are and where you're going, and pick the appropriate sign, as needed.



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