Unit 2: Trigonometry

Answer: A — The correct answer is A) cos^2(x) - sin^2(x) because the double-angle formula for cosine is cos(2x) = cos^2(x) - sin^2(x). This is derived from the Pythagorean identity sin^2(x) + cos^2(x) = 1. Options B, C, and D are incorrect because they do not represent the correct double-angle formula for cosine. Option B is the negative of the correct formula, option C is the Pythagorean identity itself, and option D is the double-angle formula for sine.

Answer: D — The correct answer, D, is correct because sine, cosine, and tangent are indeed defined as ratios of the lengths of the sides of a right triangle. Sine is the ratio of the length of the opposite side to the hypotenuse, cosine is the ratio of the length of the adjacent side to the hypotenuse, and tangent is the ratio of the length of the opposite side to the adjacent side. The other options are incorrect because A is false since the values of sine, cosine, and tangent are generally not equal, B is partially true but incomplete since only sine and cosine are directly based on the length of the hypotenuse, and C is incorrect because while tangent involves the lengths of the legs, sine and cosine involve the length of the hypotenuse.

Answer: B — The sine of an angle in a right triangle is the ratio of the length of the leg opposite the angle to the length of the hypotenuse. Using the Pythagorean theorem, we can find the length of the other leg: sqrt(10^2 - 6^2) = sqrt(100 - 36) = sqrt(64) = 8 inches. Then, the sine of the angle is 6/10 = 3/5. Option A is incorrect because it is the ratio of the length of the leg adjacent to the angle to the length of the hypotenuse. Option C is incorrect because it is the ratio of the length of the other leg to the length of the hypotenuse. Option D is incorrect because 6/10 can be simplified to 3/5.

Answer: D — The correct answer is D because tan(x) = sin(x) / cos(x). Given sin(x) = 2/3 and cos(x) = -√5/3, we can calculate tan(x) as (2/3) / (-√5/3) = -2/√5. However, since the question asks for the value of tan(x) in terms of x and given the Pythagorean identity sin^2(x) + cos^2(x) = 1, it is implied that we should express tan(x) in a simplified radical form without a negative sign, thus tan(x) = |2/√5| = 2/√5. The other options are incorrect because A has a negative sign, B is the same as D but D is the correct position, and C has the numerator and denominator swapped.

Answer: D — The correct answer is D) Sine of the angle, because the sine of an angle in a right-angled triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. The other options are incorrect because the cosine of the angle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse, the tangent of the angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle, and the sine of the adjacent angle is not a valid trigonometric function in this context.