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    Trigonometry

    Solved Problems | Topics

    Proving Identities
    ✔Show that: \(1 + {\left( {\cot \theta } \right)^2} = {\left( {\csc \theta } \right)^2}\)
    ✔\(\frac{{1 + {{\tan }^2}\theta }}{{\csc \theta }} = \sec \theta \tan \theta \)
    ✔\(\tan \theta + \cot \theta = \sec \theta \csc \theta \)
    Angle of Elevation and Depression
    ✔A railroad is inclined at an angle of \(50'\) with the horizontal. How many feet does it rise in a horizontal distance of 2 miles?
    ✔A flagpole and a tower stand 36 meters apart on a horizontal plane. A person standing successively at their bases observes that the angle of elevation of the top of the tower is twice that of the pole, but at a point midway between their bases and angles of elevation are complementary. Find the height of the pole and tower.
    ✔At a point A south of a tower the angle of elevation of the top of the tower is 50 o. At another point B 200 meters east of A, the angle of elevation is 22o. Find the height of the tower.
    Right Triangles
    ✔A flagpole is 34 feet high stands on top of a tower 30 feet high. From a certain point in the same horizontal plane with the base of the tower, the angle subtended by the pole is equal to the angle of elevation of the top of the tower. Find the distance from this point to the base of the tower.
    ✔A flagpole is 34 feet high stands on top of a tower 30 feet high. From a certain point in the same horizontal plane with the base of the tower, the angle subtended by the pole is equal to the angle of elevation of the top of the tower. Find the distance from this point to the base of the tower.
    ✔A balloon is rising at the rate of 10 ft a second and is being carried horizontally by a wind which has a velocity of 15 miles an hour. Find the actual velocity and the angle that its path makes with the vertical.
    ✔A flagpole broken over by the wind forms a right triangle with the ground. If the angle which the broken part makes with the ground. If the angle which the broken part makes with the ground is 50 degrees and the distance from the tip of the pole to the foot is 55 feet, how tall was the pole?
    ✔A wire is stretched from the top of a vertical pole standing on the level ground. The wire reaches to a point on the ground 10 feet from the foot of the pole and makes an angle of 75o with the horizontal. Find the height of the pole and the length of the wire.
    Oblique Triangles
    ✔Points \(A\) and \(B\) are separated by an obstacle. In order to find the distance between the, a third point \(C\) is selected which is \(120 ~ yards\) from \(A\) and \(150~yards\) from \(B\). The angle \(ABC\) is measured to be \(80^o10'\). Find the distance from \(A\) to \(B\).
    ✔A military observer notes two enemy batteries which subtend, at his observation post, an angle of \(40^o\). The interval between the flash and the report of a gun is 5 seconds for one battery, and 4 seconds for the other. If the velocity of sound is 1140 ft a second, how far apart are the batteries?
    ✔The diagonals of a parallelogram are 7 inches and 9 inches respectively, they intersect at an angle of \(52^o\). Find the sides of the parallelogram.
    ✔Two ships leave a dock at the same time. One sail northeast at the rate of 8.5 miles an hour, the other sails north at the rate of 10 miles an hour. How far apart are they at the end of 2 hours.
    ✔Three circles of radii 3, 4 and 5 inches respectively are tangent to each other externally. Find the angles of the triangle formed by joining the centers.
    Circles and Triangles
    ✔The area of a rectangular pentagon is 560 square ft. Find the radii of the circumscribed and inscribed circles.
    ✔The radius of a circle is 40 inches, the length of the chord is 70 inches. Find the central angle subtended by the chord.
    Trigonometric Equations
    ✔Show that the area of a regular polygon of \(n\) sides circumscribed about a circle of radius r is \(nr^2 \tan \frac{180^o}{n}\)
    ✔The speed of light in ethyl alcohol is approximately 220, 400 km/sec. A light ray leaves a point in air and strikes a point in a jar of ethyl alcohol with an angle of refraction of 38 degrees. To the nearest minute, what is the angle of incidence?
    Equations
    ✔Find the distance between points A and B, defined by the points: A(cos A, -sin A) and B (sin A, Cos A) .

    Tags: Trigonometry, Trigonometry Problems, Trigonometry worded problems, Trigonometry applications, trigonometric functions, Trigonometry basic formulas, Complementary angles, angles, angle of depression and angle of elevation, trigonometry formula, plane trigonometry, spherical trigonometry

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