An angle in the first quadrant has a sine of \displaystyle \frac{1}{\sqrt{2}} . Find the tangent of the same angle.

A 25-m-high pine tree is growing in soft ground. After a storm, the tree leans at an angle of 60° with the ground. A pine cone falls from the top of the tree to the ground. Determine an expression for the exact distance that the pine cone falls.

A. \displaystyle 25 \sqrt{3} m

B. \displaystyle \frac{25}{2} m

C. \displaystyle \frac{\sqrt{3}}{2} m

D. \displaystyle \frac{25\sqrt{3}}{2} m

An angle 6 in the third quadrant has a sine of -0.6133. In which quadrant is another angle with the same sine?

An angle in the second quadrant has a tangent of \displaystyle -\frac{3}{4} . Another angle with the same tangent measures about

A. 37^o

B. 53^o

C. 127^o

D. 323^o

In \triangle PQR, \angle P = 25^o, \angle R = 65^o, and q =12 cm. To determine the length of p, what is the most appropriate trigonometric tool?

A. the sine law

B. the cosine law

C. primary trigonometric ratios

D. reciprocal trigonometric ratios

a) \angle A lies in the second quadrant and has a cotangent of \displaystyle -\frac{5}{7} . Sketch a diagram showing the position of \angle A, including a triangle with the lengths of the sides labelled.

b) Determine expressions for the other five trigonometric ratios for \angle A.

A hot-air balloon is used to give rides to visitors at a summer fair. The balloon is tethered to the ground by a long cable. The cable is extended to its maximum length of 300 m, and the wind is blowing the balloon such that the cable makes an angle of 60° with the ground. The cable is pulled in to 200 m, but the wind strengthens, decreasing the angle to 45°.

a) Sketch the two positions of the balloon, including distances and angles.

b) Find an exact expression for the horizontal distance that the balloon moves between the two positions.

Tanis leaves home and rides her bicycle 12 km north. She turns east and rides another 5 km. Then, she turns onto a forest bicycle path that runs 45° south of east and rides for another 5 km.

a) Sketch a diagram of Tanis’s journey.

b) What is the most appropriate trigonometric tool to use in determining her distance from home? Justify your answer.

c) How far is Tanis from home at this point?

d) Which direction will take her directly home?

Antonio parks his car in the parking lot of a mountain-biking area and then bikes 1.4 km along a level trail in a direction 20° east of north. He turns right through an angle of 140° and rides up a sloping trail with an angle of elevation of 15°. When Antonio reaches the top of a hill, his odometer indicates that he has come another 1.2 km. He notices that there is a path sloping downward directly back to the parking lot. Antonio takes this path, returning to his car. Determine the total distance that he rode his bike, to the nearest tenth of a kilometre.

While visiting relatives in the Azores Islands, Juan sails from their home on Sao Jorge to Faial, a 45-km ferry ride. From Faial, he measures the angle of elevation to the Pico volcano as 5.8° and the angle of separation between the base of the volcano and Sao Jorge as 80°. When he returns to Sao Jorge, Juan measures the angle of separation of Faial and the base of the Pico volcano as 29°. Use this information to determine the height of the volcano, to the nearest metre.

Use a unit circle to show that \cos \theta = \cos (360^o - \theta) is an identity.

Prove that \frac{\csc \theta}{\sec \theta} = \cot \theta

Prove that \displaystyle \frac{\sin \theta }{1 - \cos \theta} = \csc \theta (1 + \cos \theta)