Trigonometry #6 - Questions and Answers

Question 1 Let θ be an angle in quadrant 3 such that sin θ = -5/13 Find the exact values of sec θ and cot θ. Answer We have to find the trigonometry ratios θ is in quadrant 3 sin θ = -5/13 cos θ = ±√(1 - sin²θ) = ±√(1 - (-5/13)²) = ±√(1 - 25/169) = ±√(144/169) = ±12/13 cos θ = -12/13 (since cos is negative in quadrant 3) sec θ = 1/cos θ = 1/(-12/13) = -13/12 = -1.083 cot θ = cos θ/sin θ = (-12/13) / (-5/13) = 12/5 = 2.4 sec θ = -13/12 = -1.083 cot θ = 12/5 = 2.4 Question 2 If sin x = 1/3, x in quadrant 1, then find sin(2x), cos(2x), tan(2x) Answer Here we solve trigonometry problem sin x = 1/3, x in quadrant 1 sin(2x) = 2sin x cos x cos x = √(1 - sin²x) = √(1 - (1/3)²) = √(1 - 1/9) = √(8/9) = 2√2/3 sin(2x) = 2 * (1/3) * (2√2/3) = 4√2/9 cos(2x) = 2cos²x - 1 = 2 * (2√2/3)² - 1 = 2 * (8/9) - 1 = 16/9 - 1 = 7/9 tan(2x) = sin(2x)/cos(2x) = (4√2/9)/(7/9) = 4√2/7 Question 3 P(x, y) is the endpoint of the unit circle determined by t. Then sint =, cost =, and tant = . Answer 1 According to the definition, sint is equivalent to y. cost is equivalent to x. tant is equivalent to y/x The unidentified terms are x, y, y/x Answer 2 Explanation: Let's solve the problem step by step: Sine of angle t (sint): We know that the sine of an angle is defined as the y-coordinate of the corresponding point on the unit circle. Therefore, sint = y. Cosine of angle t (cost): Similarly, the cosine of an angle is defined as the x-coordinate of the corresponding point on the unit circle. Hence, cost = x. Tangent of angle t (tant): Tangent is the ratio of sine to cosine, so tant = sint/cost Answer 3 Let's assume that the angle t is measured counterclockwise from the positive x-axis to the line segment connecting the origin to the point P(x, y). To find sin t, we can use the y-coordinate of the point P. Since P lies on the unit circle, the y- coordinate is equal to sin t. Therefore, sint = y. To find cos t, we can use the x-coordinate of the point P. Similarly, since P lies on the unit circle, the x-coordinate is equal to cos t. Therefore, cost = x. To find tan t, we can use the x-coordinate of the point P. Similarly, since P lies on the unit circle, the x-coordinate is equal to cos t. Therefore, cost = x. Finally, to find tan t, we can divide the y-coordinate by the x-coordinate, as tan t is defined as the ratio of sin t to cos t. Therefore, tant = y/x. These are the solutions for sin t, cos t, and tan t based on the given endpoint P(x, y) on the unit circle. Answer 4 Step 1: To solve the problem, we need to find the values of sint, cost, and tant given that P(x,y) is the endpoint of the unit circle determined by t. In a unit circle, the coordinates of a point on the circle can be expressed as (cost, sint), where t is the angle measured from the positive x-axis to the terminal side of the angle. Step 2: Therefore, in this case, we have P(x, y) = (cost, sint). Hence, we can determine the values of sint, cost, and tant as follows: sint = y cost = x tant = sint/cost = y/x Thus, the values of sint, cost, and tant for the endpoint P(x,y) of the unit circle determined by t are sint = y, cost = x, and tant = y/x. Question 4 Evaluate the logarithmic expression without using a calculator. log 100,000. Answer 1 100000 = 10⁵ Apply the basic property of the base -10 logarithm: log 10⁵ = 5 Answer 2 WE REWRITE THE ARGUMENT AS: 100000 = 10⁵ •WE APPLY THE PROPERTY OF THE BASE -10 LOGARITHM: LOG 10⁵ = 5 Question 5 sin(theta) is opposite/hypotenuse regarding right triangles. Then what would sin (2θ) be? Answer 1 The number sin 2θ is the sin of twice the angle θ. It is almost never equal to 2sin θ. sin 2θ = 2sin θ x cos θ (1) If you know that sin θ = x/3, all you need to do to find sin 2θ is to find cos θ. Answer 2 sin 2θ ≠ 2sin θ, but 4sin 2θ = 2sin θcos θ As we know, cos 2θ≠ 2 cos θ, cos 2θ = cos²θ- sin²θ = 2 cos²θ-1 = 1 - 2 sin²θ