Download (b) the right angle

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