Trigonometry

A train is moving at the rate of 8 mi/h along a piece of circular track of radius 2,500 ft. Through what angle does it turn in 1 min?

An artist wishes to make a sign in the shape of an isosceles triangle with a 42 degree vertex angle and a base of 18 m. What is the area of the sign?

A 20-ft lamp post casts a 25 ft shadow. At the same time, a nearby building casts a 50 ft shadow. How tall is the building?

Three circles of radii 3, 4 and 5 inches, respectively, are tangent to each other externally. Find the largest angle of the triangle found by joining the centers of the circles.

Simplify the expression \(\cos^2\theta - \sin^2\theta\).

\(\csc 520^\circ = ?\)

Simplify \(\dfrac{\cos A}{1 - \sin A} - \tan A\).

If the tangent of angle A is equal to the square root of 3, with angle A in the 3rd quadrant, find the square of the tangent of A/2.

At a height of 23,240 ft, a pilot of an airplane measures the angle of depression of a light at an airport as 28°45′. How far is he from the light?

A tower 150 m high is situated at the top of a hill. At a point 650 m down the hill, the angle between the surface of the hill and the line of sight to the top of the tower is 12°30′. Find the inclination of the hill to a horizontal plane.

What is the angle of the sun above the horizon when a building 150 ft high casts a shadow of 405 ft?

Simplify \(\cos^2 x + \sin^2 x + \tan^2 x\).

Find the exact value of \(\dfrac{\tan 25^\circ + \tan 50^\circ}{1 - \tan 25^\circ\tan 50^\circ}\).

What is the value of x in \(\arctan 3x + \arctan 2x = 45^\circ\)?

From a hill 600 ft high, the angles of depression to the bases in opposite directions are 42° and 19°23′ respectively. Find the length of the proposed tunnel through the bases.

Two buildings with flat roofs are 60 m apart. From the roof of the shorter building, 40 m in height, the angle of elevation to the edge of the roof of the taller building is 40°. How high is the taller building?

Three ships are situated as follows: A is 225 mi due north of C, and B is 375 mi due east of C. What is the bearing of B from A?

A support wire is anchored 12 m up from the base of a flagpole and the wire makes a 15° angle with the ground. How long is the wire?

The base of an isosceles triangle is 20.4 and the base angles are 48°40′. Find the altitude of the triangle.

Find the exact value of \(\sec\left(-\dfrac{\pi}{6}\right)\).