Questions: Trigonometric identities (degrees)

Author

Dzhemma Ruseva

Summary

A selection of questions on trigonometric identities, where angles are measured in degrees.

Before attempting these questions, it is recommended that you read Guide: Trigonometric identities (degrees).

Q1

Using trigonometric identities, find the values of the following expressions:

1.1. \(\quad2(6\sin^2(\theta))+3(4\cos^2(\theta))\).

1.2. \(\quad10(7\sin^2(\theta))+14(5\cos^2(\theta))\).

1.3. \(\quad5\left(\dfrac{6}{\csc^2(\theta)}\right)+15\left(\dfrac{2}{\sec^2(\theta)}\right)\).

1.4. \(\quad(\cos^2(\theta)-\sin^2(\theta))^2 + 4sin^2(\theta)\cos^2(\theta)\)

1.5. \(\quad2\sin(30)\cos(15)+2\cos(30)\sin(15)\)

1.6. \(\quad3\cos(45)\cos(15)-3\sin(45)\sin(15)\)

1.7. \(\quad\sin(150)+\sin(30)\)

1.8. \(\quad\cos(150)+\cos(30)\)

Simplify the following expressions:

2.1. \(\quad\tan(\theta)\cos(-\theta)\)

2.2 \(\quad\tan(-\theta)\csc(-\theta)\sec(-\theta)\)

2.3. \(\quad\tan^2(\theta)+\sin^2(\theta)+\cos^2(\theta)\)

2.4. \(\quad\displaystyle\frac{2\sin(\theta)}{\cos(\theta)(1-\tan^2(\theta))}\)

2.5. \(\quad\displaystyle\frac{\sin(7\theta)+\sin(3\theta)}{\cos(7\theta)-\cos(3\theta)}\)

2.6. \(\quad\displaystyle\frac{\sin(5\theta)-\sin(\theta)}{\cos(5\theta)+\cos(\theta)}\)

Using trigonometric identities, answer the following questions:

3.1. What is the value of \(\cos(210^\circ)\)?

3.2. What are the values of \(\sin(135^\circ)\) and \(\sin(225^\circ)\)?

3.3. If \(\sin(50^\circ)\) has the value \(0.766\) (to \(3\) decimal places), what is the value of \(\cos(130^\circ)\) to three decimal places?

Using trigonometric identities, find exact values of the following:

4.1. \(\quad\sin(15)\)

4.2. \(\quad\cos(15)\)

4.3. \(\quad\tan(15)\)

4.4. \(\quad\sin(75)\)

4.5. \(\quad\cos(75)\)

4.6. \(\quad\tan(75)\)

v1.0: initial version created 08/23 by Dzhemma Ruseva as part of a University of St Andrews STEP project.

v1.1: edited 05/24 by tdhc, and split into versions for both degrees and radians.