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Example of trigonometry function of any angles

Question

a is the angle of quadrant2, cos 2a = -5/13. Find the value of tan (pi/4 - 2a)

Solution

Because a is the angle of quadrant2, so, the terminal side of the angle a lies in quadrant2. then k × 2pi + pi/2 < a < pi + k × 2pi. Multiply 2 to the iequility, k × 4pi + pi < 2a < 2pi + k × 4pi. So, 2a either in quadrant3 or in quadrant4. Because we are given cos 2a = -5/13. That is, the value of cos 2a is negative. Because the value of cosine of an angle of quadrant4 should be positive, so 2a is an angle of quadrant3. In quadrant3, the value of sine of an angle is positive. Using the formula, sin2 2a + cos2 2a = 1. So, sin2 2a = 1 - cos2 2a = 1 - (-5/13)2 = 1 - 25/169 = 169/169 - 25/169 = 144/169. sin 2a = - square root of 144/169 = -12/13. tan 2a = sin 2a/cos 2a = -12/13 ÷ -5/13 = -12/13 × -13/5 = 12/5. Using the formula, tan (pi/4 - 2a) = (tan pi/4 - tan 2a)/[1 + tan pi/4 × tan 2a] = (1 - 12/5)/[1 + (1)(12/5)] = (1 - 12/5)/(1 + 12/5) = (5 - 12)/5 ÷ (5 + 12)/5 = (5 - 12)/5 × 5/(5 + 12) = -7/17.