An observer stands on one side of a river and spots a tree directly across on the other side. The observer walks 100 meters downstream and finds that the angle formed by the tree and his original position is 30 degrees. What is the width of the river?

A) 57.74

B) 62.32

C) 51.04

D) 69.99

Verify the identity: \$ ( 1 + tanθ )^2 = ( 1 + 2 sinθ cosθ)/(cos^2 θ) \$.

A) 0

B) Proved

C) Not proved

D) 1

Given that \$sinx = 3/5\$ and x is in the first quadrant, find \$tan(90^\circ - x)\$.

A) \$ 4/5\$

B) \$ 3/5\$

C) \$ 4/3\$

D) \$ 3/4\$

A private plane flies 1.3 hours at 110 mph on a bearing of 40°. Then it turns and continues another 1.5 hours at the same speed, but on a bearing of 130°. At the end of this time, how far is the plane from its starting point? What is its bearing from that starting point? Round your answers to whole numbers.

A) Distance = 212 miles and Angle = 71 degrees

B) Distance = 118 miles and Angle = 80 degrees

C) Distance = 200 miles and Angle = 90 degrees

D) Distance = 218 miles and Angle = 89 degrees

If \$tan θ + sec θ = 2/\sqrt3\$ and θ is a positive acute angle, find the value of sin θ.

A) \$ 1/3\$

B) \$ 5/2\$

C) \$ 1/2\$

D) \$ 1/7\$

Find the value of the product of the six trigonometric functions.

A) 2

B) 3

C) 1

D) 4

Simplify the expression: \$ (sin^2(- θ) - cos^2(- θ)) / (sin(- θ) - cos (- θ))\$.

A) tanθ - cotθ

B) cosθ - sinθ

C) tanθ + cotθ

D) cosθ + sinθ

If \$x sin^3(θ) + y cos^3 (θ) = sin θ cos θ\$ and x sin θ – y cos θ = 0, then prove that \$x^2 + y^2 = 1\$, (where, sin θ ≠ 0 and cos θ ≠ 0).

A) Proved

B) 0

C) Not proved

D) Infinity

A flagpole casts a shadow that is 25 meters long when the angle of elevation of the sun is 30°. What is the height of the flagpole?

A) 40.43 m

B) 14.43 m

C) 10.43 m

D) 41.43 m

If \$cos⁡(θ) = 5/13\$​ and θ is in the fourth quadrant, find tan⁡(θ).

A) \$(12)/5\$

B) \$- 5/(12)\$

C) \$- (12)/5\$

D) \$5/(12)\$