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Evaluate each expression: (i) tan(arccos(8/17)); (ii) cos(arctan(3/4)); (iii) sin(arccos(3/5)); (iv) tan(arccos(1/3)); (v) cosec(arcsec(sqrt(41)/5)); (vi) tan(arccosec(65/63)).

1. (i) 15/8 (ii) 4/5 (iii) 4/5 (iv) 2*sqrt(2) (v) sqrt(41)/4 (vi) 63/16
2. (i) 8/15 (ii) 3/5 (iii) 3/5 (iv) sqrt(2) (v) sqrt(41)/5 (vi) 16/63
3. (i) 15/8 (ii) 4/5 (iii) 3/5 (iv) 2*sqrt(2) (v) 4/sqrt(41) (vi) 16/63
4. (i) 17/8 (ii) 4/5 (iii) 4/5 (iv) 3 (v) sqrt(41)/4 (vi) 63/65

Correct answer: (i) 15/8 (ii) 4/5 (iii) 4/5 (iv) 2*sqrt(2) (v) sqrt(41)/4 (vi) 63/16

Solution

(i) cos = 8/17 -> opposite = sqrt(17² - 8²) = 15, tan = 15/8. (ii) tan = 3/4 -> hyp = 5, cos = 4/5. (iii) cos = 3/5 -> opp = 4, sin = 4/5. (iv) cos = 1/3 -> opp = sqrt(9-1) = sqrt(8) = 2*sqrt(2), tan = 2*sqrt(2)/1 = 2*sqrt(2). (v) sec = sqrt(41)/5 -> hyp = sqrt(41), adj = 5, opp = sqrt(41-25) = 4, cosec = hyp/opp = sqrt(41)/4. (vi) cosec = 65/63 -> hyp = 65, opp = 63, adj = sqrt(65² - 63²) = sqrt(4225 - 3969) = sqrt(256) = 16, tan = opp/adj = 63/16.

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