Question - Trigonometric Ratios in a Right-Angled Triangle
Solution:
The image shows a right-angled triangle, ΔABC, with angle A = 60°, angle B = 90°, and the hypotenuse AC = 20 ft. We are required to solve a series of questions related to this triangle.a) Using trigonometric ratios to find the perpendicular (opposite side to angle A, which is BC) and base (adjacent side to angle A, which is AB):We use the sine and cosine functions, which are defined as follows for a right-angled triangle:- Sine of an angle is the ratio of the length of the opposite side to the hypotenuse.- Cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.For angle A,sin(60°) = opposite/hypotenuse => sin(60°) = BC/ACBC = AC * sin(60°)BC = 20 * (√3/2) (since sin(60°) is √3/2)BC = 10√3 ftFor the base (AB),cos(60°) = adjacent/hypotenuse => cos(60°) = AB/ACAB = AC * cos(60°)AB = 20 * (1/2) (since cos(60°) is 1/2)AB = 10 ftb) Using the perpendicular (BC) and base (AB) obtained above, find the value of tan(60°):Tangent of an angle is the ratio of the length of the opposite side to the adjacent side.tan(60°) = opposite/adjacenttan(60°) = BC/ABtan(60°) = (10√3)/10tan(60°) = √3c) Find the values of Sin C and Cos C:In a right triangle, the sine of one non-right angle is the cosine of the other, and vice versa.Since angle C is the 90-angle B (90° - 60° = 30°), we can find the sine and cosine of angle C by using their known values at 30°.sin(C) = sin(30°) = 1/2cos(C) = cos(30°) = √3/2d) Using the values of Sin C and Cos C, prove that Sin² C + Cos² C = 1:This is a well-known trigonometric identity known as the Pythagorean identity.Now let's substitute the values obtained for Sin C and Cos C:Sin² C + Cos² C = (1/2)² + (√3/2)²Sin² C + Cos² C = 1/4 + 3/4Sin² C + Cos² C = 4/4Sin² C + Cos² C = 1This proves the Pythagorean identity for angle C.
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