Questions: Trigonometric identities (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. \(\quad\cos(150)+\cos(30)\)
Q2
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. \(\quad\displaystyle\frac{\sin(5\theta)-\sin(\theta)}{\cos(5\theta)+\cos(\theta)}\)
Q3
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?
Q4
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. \(\quad\tan(75)\)
After attempting the questions above, please click this link to find the answers.
Version history and licensing
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.