Questions: Trigonometric identities (radians)
Before attempting these questions, it is recommended that you read Guide: Trigonometric identities (radians).
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. \(\quad2\sin(\pi/6)\cos(\pi/12)+2\cos(\pi/6)\sin(\pi/12)\)
1.6. \(\quad3\cos(\pi/4)\cos(\pi/12)-3\sin(\pi/4)\sin(\pi/12)\)
1.7. \(\quad\sin(5\pi/6)+\sin(\pi/6)\)
1.8. \(\quad\cos(5\pi/6)+\cos(\pi/6)\)
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 \(\cos(-7\pi/6)\)?
3.2. What are the values of \(\sin(3\pi/4)\) and \(\sin(5\pi/4)\)?
3.3. If \(\sin(5\pi/18)\) has the value \(0.766\) (to \(3\) decimal places), what is the value of \(\cos(13\pi/18)\) to three decimal places?
Q4
Using trigonometric identities find exact values of the following:
3.1. \(\quad\sin(\pi/12)\)
3.2. \(\quad\cos(\pi/12)\)
3.3. \(\quad\tan(\pi/12)\)
3.4. \(\quad\sin(5\pi/12)\)
3.5. \(\quad\cos(5\pi/12)\)
3.6. \(\quad\tan(5\pi/12)\)
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.