Basic Trigonometry Problems: Problems with Solutions

Basic trigonometry problems: degrees and radians, signs of the trigonometric functions by quadrant, values for any angle with the reduction formulas, finding the other functions from one of them, and right-triangle problems solved with trigonometric ratios. Step-by-step solutions included.

Problem 1

Express in radians the measure of an angle of [tex]735^{\circ}[/tex].

[tex]\frac{47\pi}{12}[/tex]

[tex]\frac{49\pi}{12}[/tex]

[tex]\frac{25\pi}{12}[/tex]

[tex]\frac{49\pi}{6}[/tex]

Problem 2

The initial side of an angle of [tex]-800^{\circ}[/tex] coincides with the positive direction of the [tex]x[/tex]-axis. In which quadrant does its terminal side lie?

[tex]\mathrm{I}[/tex]

[tex]\mathrm{II}[/tex]

[tex]\mathrm{III}[/tex]

[tex]\mathrm{IV}[/tex]

Problem 3

Which of the following numbers is negative?

[tex]\cot 115^{\circ}[/tex]

[tex]\cos 340^{\circ}[/tex]

[tex]\cot 195^{\circ}[/tex]

[tex]\tan 200^{\circ}[/tex]

Problem 4

The points [tex]A[/tex] and [tex]B[/tex] lie on a circle with centre [tex]O[/tex] and radius [tex]2[/tex] cm, and the length of the smaller arc [tex]AB[/tex] is [tex]2. 5[/tex] cm. Find the radian measure of [tex]\angle AOB[/tex].

[tex]5[/tex]

[tex]\frac{4}{5}[/tex]

[tex]\frac{5}{4}[/tex]

[tex]\frac{5}{8}[/tex]

Problem 5

Calculate [tex]\sin 120^{\circ}\times \cot 150^{\circ}+\sin 60^{\circ}\times \tan 150^{\circ}[/tex].

Problem 6

Calculate [tex]\sin 120^{\circ}+\cos 1125^{\circ}+\tan 750^{\circ}[/tex].

[tex]\frac{5\sqrt{3}-3\sqrt{2}}{6}[/tex]

[tex]\frac{3\sqrt{2}-\sqrt{3}}{6}[/tex]

[tex]\frac{3\sqrt{3}+\sqrt{2}}{2}[/tex]

[tex]\frac{5\sqrt{3}+3\sqrt{2}}{6}[/tex]

Problem 7

One nautical mile equals the length of an arc of the Earth's equator corresponding to a central angle of [tex]1'[/tex] (one minute of arc). Take the radius of the Earth to be [tex]6370[/tex] km. About how many kilometres is one nautical mile?

[tex]\approx 111. 2[/tex]

[tex]\approx 1. 853[/tex]

[tex]\approx 3. 706[/tex]

[tex]\approx 0. 031[/tex]

Problem 8

The terminal side of the angle [tex]\alpha[/tex] lies in the third quadrant and [tex]\cos(-\alpha)=-\frac{1}{3}[/tex]. Find [tex]\sin \alpha[/tex] and [tex]\tan \alpha[/tex].

[tex]\sin \alpha=\frac{2\sqrt{2}}{3}, \ \tan \alpha=-2\sqrt{2}[/tex]

[tex]\sin \alpha=-\frac{2\sqrt{2}}{3}, \ \tan \alpha=-2\sqrt{2}[/tex]

[tex]\sin \alpha=-\frac{2\sqrt{2}}{3}, \ \tan \alpha=2\sqrt{2}[/tex]

[tex]\sin \alpha=-\frac{\sqrt{6}}{3}, \ \tan \alpha=\sqrt{6}[/tex]

Problem 9

Find [tex]\tan \alpha[/tex] if [tex]\frac{7\sin \alpha+13\cos\alpha}{5\sin \alpha-17\cos\alpha}=3[/tex].

Problem 10

Find the distance from the vertex [tex]A[/tex] to the line [tex]BC[/tex] in [tex]\triangle ABC[/tex] if [tex]AC=5[/tex] cm, [tex]BC=8[/tex] cm and [tex]\angle ACB=60^{\circ}[/tex].

[tex]\frac{5\sqrt{3}}{2}[/tex]

[tex]\frac{5}{2}[/tex]

[tex]4\sqrt{3}[/tex]

[tex]7[/tex]

Problem 11

[tex]CD[/tex] is the altitude to the hypotenuse of the right triangle [tex]ABC[/tex] ([tex]\angle ACB=90^{\circ}[/tex]). If [tex]BC=4\sqrt{3}[/tex] cm and [tex]\cos\angle BAC=\frac{\sqrt{3}}{4}[/tex], find the length of [tex]CD[/tex] in centimetres.

Problem 12

The area of a right triangle is [tex]30[/tex] cm², and the tangent of one of its acute angles is [tex]2. 4[/tex]. Find the hypotenuse of the triangle in centimetres.

Problem 13

The perimeter of an isosceles triangle is [tex]64[/tex] cm, and the cosine of its base angle is [tex]0. 28[/tex]. Find the altitude to the base in centimetres.

Problem 14

Calculate the sum [tex]\sin 110^{\circ}+\sin 130^{\circ}+\sin 150^{\circ}+\dots+\sin 250^{\circ}+\sin 270^{\circ}[/tex], in which the angles increase by [tex]20^{\circ}[/tex].

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Basic Trigonometry Problems: Problems with Solutions

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