Trigonometric equations

Express 2 sin x + 5 cos x in the form R sin (x + α)

where R > 0 and 0° ≤ α ≤ 90°

In 2010, the water temperature at a beach could be modelled by the formula

T = 14.2 – (2 sin d ° + 5 cos d °)

where: T is the temperature of the water in °C d is the number of days after 1 January 2010

Find the value of d when the temperature of the water reached its lowest value in 2010

Find the maximum temperature of the water predicted by the model in 2010

Give your answer to one decimal place.

Find the number of weeks in 2010 for which the temperature of the water was higher than 15 °C

Give your answer to the nearest whole number.

Show that the expression

sin 2 θ cosec θ + cos 2 θ sec θ

can be written as

4 cos θ – sec θ

where sinθ ≠ 0 and cosθ ≠ 0

Explain why the student should reject one of their values for cosθ in Step 4.

State the correct solutions to the equation

Given that

show that

Hence, solve

in the interval 0 2

Give your answers to two decimal places.

Solve

Give your answers to one decimal place.

Show that the equation

can be written in the form

Hence, given x is obtuse and

find the exact value of tan x

Fully justify your answer.

p(x) 30x³ 7x² 7x 2

Prove that (2x 1) is a factor of p(x)

Factorise p (x) completely.

Prove that there are no real solutions to the equation