Prove that \displaystyle
\frac{1-2\sin^2x}{\cos x + \sin x} + 2\sin \frac{x}{2}\cos \frac{x}{2} = \cos x
.
Solve for x. \displaystyle
\cos 2x + 2\sin^2x - 3 =-2
, where 0\leq x \leq 2\pi
Determine the solution(s) for the of the following equation, where 0\leq x \leq 2\pi
.
\displaystyle
\cos x = \frac{\sqrt{3}}{2}
Determine the solution(s) for the of the following equation, where 0\leq x \leq 2\pi
.
\displaystyle
\tan x = - \sqrt{3}
Determine the solution(s) for the of the following equation, where 0\leq x \leq 2\pi
.
\displaystyle
\sin x = - \frac{\sqrt{2}}{2}
The quadratic trigonometric equation a \cos2 x +b\cos x - 1 = 0
has the solutions \frac{\pi}{3}, \pi
, and \frac{5\pi}{3}
in the interval 0 \leq x \leq 2\pi
. What are the values of a
and b
?
The depth of the ocean at a swim buoy can be modelled by the
function d(t) = 4 + 2\sin(\frac{\pi}{6}t)
, where d
is the depth of water in metres and t is the time in hours, if 0 \leq t \leq 24
. Consider a day when t = 0
represents midnight. Determine when the depth of water is 3 m.
Nina needs to find the cosine of \frac{11\pi}{4}
. If she knows the sine and cosine of \pi
, as well as the sine and cosine of \frac{7\pi}{4}
, how can she find the cosine of \frac{11\pi}{4}
? What is her answer?
Solve 3 \sin x + 2 = 1.5
, where 0\leq x \leq 2\pi
.
The tangent of the acute angle a is 0.75
, and the tangent of the acute angle \beta
is 2.4
. Without using a calculator, determine the value of
\sin (\alpha - \beta)
and \cos(\alpha + \beta)
.
The angle x
lies in the interval \frac{\pi}{2} \leq x \leq \pi
, and \sin^2x = \frac{4}{9}
. Determine the value of the following. State your answer in exact values.
\displaystyle
\sin 2x
The angle x
lies in the interval \frac{\pi}{2} \leq x \leq \pi
, and \sin^2x = \frac{4}{9}
. Determine the value of the following. State your answer in exact values.
\displaystyle
\cos 2x
The angle x
lies in the interval \frac{\pi}{2} \leq x \leq \pi
, and \sin^2x = \frac{4}{9}
. Determine the value of the following. State your answer in exact values.
\displaystyle
\cos \frac{x}{2}
The angle x
lies in the interval \frac{\pi}{2} \leq x \leq \pi
, and \sin^2x = \frac{4}{9}
. Determine the value of the following. State your answer in exact values.
\displaystyle
\sin 3x
Solve for x
when -2\pi \leq x \leq 2\pi
.
\displaystyle
2-14\cos x = -5
Solve for x
when -2\pi \leq x \leq 2\pi
.
\displaystyle
9-22\cos x - 1= 19
Solve for x
when -2\pi \leq x \leq 2\pi
.
\displaystyle
2 + 7.5 \cos x = -5.5