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WUT FOUNDATION YEAR – summer 2016/2017 angles & trigonometry T.1. Draw: (a) an acute angle, (b) the right angle, (c) an obtuse angle, (d) the straight angle, (e) a concave angle, (f) the full angle. T.2. Express as radians: (a) 360◦ , T.3. Express as degrees: (a) 5π , 2 (b) 90◦ , (b) 3π , 2 (c) 60◦ , (c) π, (d) 18◦ , (d) π , 4 T.4. Find values of sin, cos, tan and cot for the following angles: (e) 15◦ , π , 6 (f) (a) α = π , 6 (e) (f) 5◦ . π . 9 (b) α = π , 3 (c) α = π . 4 T.5. Let α be an acute angle, i.e. α ∈ 0, π2 , and suppose that sin α = 12 . Compute: π π 13 (a) sin(π − α), (b) cos +α , (c) sin −α , (d) cos(2π − α), 2 2 (e) tan(π + α), (f) cot(π − α) (g) sin(2α), (h) cos(2α). Repeat the same exercise for the acute angle α such that cos α = 4 . 5 T.6. In each question draw a picture and provide appropriate calculations to answer the given problem. (a) In a right angled triangle sinus of one of the acute angles equals length 6. Find area of the angle. 1 3 and hypotenuse has (b) In a rectangle diagonal has length d and tangent of the angle between the diameter and one of the sides equals 12 . Find perimeter of the rectangle. (c) In a rectangle diagonal has length 4 and the obtuse angle between diagonals has measure Find area of the rectangle. T.7. Prove the identities: 1 + cos x cos x = , (a) tan2 x · 1 − cos x cos x (c) cos(2x) (1 + tan x tan(2x)) = 1, cos(2x) cos x − sin x = , 1 + sin(2x) cos x + sin x 1 2 cos x 1 (d) − = . 1 − cos x 1 + cos x sin2 x (b) T.8. Simplify: q (a) sin2 x(1 + cot x) + cos2 x(1 − tan x), (b) sin x − p cot2 x − cos2 x for x ∈ (π, 2π). T.9. Knowing that sin(x + y) = sin x cos y + cos x sin y and sin x + sin y = 2 sin x+y 2 find formulas for (a) cos(x + y), (b) cos x + cos y, (c) cos x − cos y. cos x−y 2 2π . 3 T.10. Find domains, ranges and periods and sketch graphs of the given functions. x π (a) f (x) = 1 − 2 sin x, (b) f (x) = cos x − , (c) f (x) = | tan(2x)|, (d) f (x) = − cot 4 3 T.11. Solve the equations: √ (a) sin x = (b) cos x = 2 ; 2 − 12 ; (c) tan(3x) = √ 3; (d) cot(2x) = −1; (e) (2 cos x − 1)(2 cos x + 1) = 3; (f) 3 sin2 x − cos2 x = 0; √ (g) tan2 x = 3 tan x; (h) 2 cos3 x + cos2 x − 2 cos x = 1; (i) sin(2x) + cos x = 0; (j) 4 cos2 x − cos(2x) = 2; (k) cos x sin(2x) = 2 sin x; √ (l) 2 sin x − sin(2x) = 3(cos x − 1); (m) tan x + cot x = 4 sin(2x). T.12. Solve the inequalities: (a) sin2 x < 1; (b) 4 cos2 x ≥ 1; (c) 3 tan2 x − 1 > 0; sin2 x 1 1 (d) ≤√ ; 3 3 (e) |2 sin x − 1| < 1; 1 < 2; (f) √ 1 − cos2 x cos(2x) (g) < 1; cos x (h) log2 cos(2x) + 2 sin x + 1 > 1. T.13. Calculate: 1 (a) arcsin , 2 (e) arcsin (−1) , (b) arccos (0) , 1 (f) arccos − , 2 (c) arctan √ 3 , (g) arctan (−1) , (d) arccot(1), √ (h) arccot − 3 . choice by Agnieszka Badeńska