Trigonometry Answers

Prove:

a. (sin𝑥+cos𝑥)^2 ≡1+2sin𝑥cos𝑥

b. (1+sin𝑥+ cos𝑥)^2 ≡2 (1+sin𝑥) (1+cos𝑥)

c. (6−cos^2𝜃)/(sin^2𝜃+5) ≡1

1.       A woman launches a boat from one shore of a straight river and wants to land at the point directly on the opposite shore. If the speed of the boat (relative to the water) is 10 mi/h and the river is flowing east at the rate of 5 mi/h, in what direction should she head the boat in order to arrive at the desired landing point?

1. Find the quadrant in which P(t) lies if sec t > 0 and csc t < 0.

2. Find the values of the 6 trigonometric functions of -7π/3.

3. Find the value of tan 7π/3 .

4. If sin t=√3/2 and cos t<0, find csc t.

5. Find the six trigonometric functions of an angle whose terminal side passes

through (-4,-3).

6. Find the remaining functions given that sin ϴ=513, sec t<0.

7. Find ϴ, if cot 2ϴ= √3, ϴ∈[0°,360°].

8. A ladder leans against the side of a building with its foot ten feet from the building. How far from the ground is the top of the ladder and how long is the ladder if it makes an angle of 60° with the ground?

9. At a certain moment Ship MV Alpha is 7.5 km. west of Ship MV Omega. Ship MV Omega is sailing S 30° 20’ E at the rate of 25 km./hr., while Ship MV Alpha is sailing directly north at rate of 20 km./hr. Find the distance between the two ships and the bearing with Ship MV Omega from Ship MV Alpha after 40 minutes.

Find the values of the 6 trigonometric functions of -(7π)/3

Find the values of the 6 trigonometric functions of tan 7π/3

Find the quadrant in which 𝑃(𝑡) lies if 𝑠𝑒𝑐 𝑡 > 0 and 𝑐𝑠𝑐 𝑡 < 0.

the angle of depression from the top of a tower to a rock on the ground is 39 degrees. If the tower is 24 m high, how far from the base of the tower is the rock?

the legs of a right triangle measure 5 inches and 7 inches

Qn 4. For a given angle θ, suppose that

sin2 θ + cos2 θ = 1 and cos2 θ − sin2 θ = cos 2θ.

2Derive an expression for cos 4θ in terms only of powers of cos θ

Qn 6. Let A, B, C and D be four points on a circle, taken in such a way that

the segments AC and BD have an intersection E. If AE = πDE, compute,

providing your working, the ratio between the areas of the triangles △AEB

and △CED, that is,

A(△AEB)

A(△CED)