Trigonometry

Evaluate the following: \(\frac{\sin^2 0^\circ + \sin^2 1^\circ + \sin^2 2^\circ + \dots + \sin^2 89^\circ + \sin^2 90^\circ}{\cos^2 0^\circ + \cos^2 1^\circ + \cos^2 2^\circ + \dots + \cos^2 89^\circ + \cos^2 90^\circ}\)

Simplify the following: \(\frac{\cos A + \cos B}{\sin A - \sin B} + \frac{\sin A + \sin B}{\cos A - \cos B}\)

Evaluate: \(\frac{2 \sin \theta \cos \theta - \cos \theta}{1 - \sin \theta + \sin^2 \theta - \cos^2 \theta}\)

Solve for the value of "A" when \(\sin A = 3.5x\) and \(\cos A = 5.5x\).

If \(\sin A = 2.511x\), \(\cos A = 3.06x\) and \(\sin 2A = 3.939x\), find the value of x?

If \(\text{coversed} \sin \theta = 0.134\), find the value of \(\theta\).

A man standing on a 48.5 meter building high, has an eyesight height of 1.5 m from the top of the building, took a depression reading from the top of another nearby building and nearest wall, which are \(50^\circ\) and \(80^\circ\) respectively. Find the height of the nearby building in meters. The man is standing at the edge of the building and both buildings lie on the same horizontal plane.

Points A and B \(1000\) m apart are plotted on a straight highway running East and West. From A, the bearing of a tower C is \(32^\circ\) W of N and from B the bearing of C is \(26^\circ\) N of E. Approximate the shortest distance of tower C to the highway.

Two triangles have equal bases. The altitude of one triangle is 3 units more than its base and the altitude of the other triangle is 3 units less than its base. Find the altitudes, if the areas of the triangles differ by 21 square units.

A ship started sailing S \(42^\circ 35'\) W at the rate of 5 kph. After 2 hours, ship B started at the same port going N \(46^\circ 20'\) W at the rate of 7 kph. After how many hours will the second ship be exactly north of ship A?

If Greenwich mean time (GMT) is 6 A.M., what is the time at a place located \(30^\circ\) East longitude?

If the longitude of Tokyo is \(139^\circ E\) and that of Manila is \(121^\circ E\) what is the time difference between Tokyo and Manila?

One degree on the equator of the earth is equivalent to

A spherical triangle ABC has an angle \(C=90^\circ\) and sides \(a=50^\circ\) and \(c=80^\circ\). Find the value of "b" in degrees.

Solve the remaining side of the spherical triangle whose given parts are \(A=B=80^\circ\) and \(a=b=89^\circ\)

Solve for side b of a right spherical triangle ABC whose parts are \(a=46^\circ\), \(c=75^\circ\) and \(C=90^\circ\).

Given a right spherical triangle whose given parts are \(a=82^\circ\) \(b=62^\circ\) and \(C=90^\circ\). What is the value of the side opposite the right angle?

Determine the value of the angle B of an isosceles spherical triangle ABC whose given parts are \(b=c=54^\circ 28'\) and \(a=92^\circ 30'\)

Solve for angle A in the spherical triangle ABC, given \(a=106^\circ 25'\) \(c=42^\circ 16'\) and \(B=114^\circ 53'\)

Solve for angle C of the oblique triangle ABC given, \(a=80^\circ\) \(c=115^\circ\) and \(A=72^\circ\).