5.3 Trigonometric Equations

Solve the equation  sin² cos for  

Given that the angle is obtuse and that  sin , find the exact value of cos .

Solve the equation tan    for  .

Solve the equation  tan sin  for  .

An isosceles triangle has sides 8 cm, 8 cm and 4 cm and equal base angles .

Find exact values for sin ,cos  and tan .

Show that    satisfies the equation  .

Hence solve the equation cos³  cos² cos   for .

A seagull sits on the surface of the sea and moves up and down as waves pass.

Its height,  metres, above its position in calm water is modelled by the function  sin  where  is the time in seconds after timing commenced.

Find the first time the seagull is 0.3 metres above its calm water position.
Give your answer to 2 decimal places.

How many times in the first minute after timing commences is the seagull 0.3 metres above its calm water position?

Show that the equation   sin² cos   can be written in the form   cos²  cos  , where ,  and  are integers to be found.

Hence, or otherwise, solve the equation  sin² cos   for  .

Given that   find the possible values of  and .

Solve the equation   sin   for  .

Solve the equation sin  cos cos for  .

A right-angled triangle has hypotenuse 8 cm.  One of its other sides is 5 cm.

Find exact values for ,  and , where  is the smallest angle in the triangle.

Show that .

Hence, or otherwise, solve the equation tan³ tan² tan  for  , giving your answers to 1 decimal place where appropriate.

A seagull sits on the surface of the sea and moves up and down as waves pass.

Its height,  metres, above its position in calm water is modelled by the function sin  where  is the time in seconds after timing commences.

Sketch a graph of  against  for  showing the coordinates of the points of intersection with the  axis.

How many times in the first minute after timing commences is the seagull 0.25 metres above its calm water position?

Find the time at which the seagull is first 0.25m above its calm water position and moving downwards. Give your answer to 3 significant figures.

Solve the equation cos² sin  for , giving your answers to 1 decimal place where appropriate.

Given that the angle is reflex and that cos   , find the exact value of tan .

Solve the equation   sin²   for  .

Solve the equation  sintan   for  , giving your answers to 1 decimal place where appropriate.

For the triangle in the diagram find exact values for sin ,cos  and tan .

Find all the values of  in the range   which satisfy the equation  tan³ tan² tan , giving your answers to 1 decimal place.

A seagull sits on the surface of the sea and moves up and down as waves pass.

Its height, metres, above its position in calm water is modelled by the function  where  is the time in seconds after timing commences.

Find the amount of time the seagull is more than 0.5 metres above its calm water position in the first 20 seconds after timing commences.

Give your answer correct to 3 significant figures.