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Trigonometry final test

  1. Convert the following from degrees to radians: [Just leave the answer in terms of \(\pi\)]

    1. \(22^\circ 30'\)
    2. \(300^\circ\)
    3. \(-75^\circ\)

  2. Convert the following from radians to degrees: [Use \(\pi=\frac{22}{7}\) where needed]

    1. \(\frac{\pi}{10}\)
    2. \(-\frac{5\pi}{4}\)
    3. \(2.2\)

  3. If you draw arcs of same length on 2 circles with radii \(15\,cm\) and \(25\,cm\) respectively, then what's the ratio of the angles made by them?

  4. If a car travels \(44\,m\) along a circular path of radius \(28\,m\), then what's the angle covered by the car on the center? [Use \(\pi=\frac{22}{7}\) where needed] [Give the answer in degrees]

  5. If \(\sec{\theta}=2\), and \(\theta\) is in \(4^{th}\) quadrant, then find all the other trigonometric ratios.

  6. Give the values of following:

    1. \(\tan{-135^\circ}\)
    2. \(\sec{405^\circ}\)
    3. \(\text{cosec}{\frac{4\pi}{3}}\)
    4. \(\sec{75^\circ}\) [Hint: Use \(\cos{(a+b)}\)]

  7. Prove the following:

    1. \(\cos{23^\circ}\cos{37^\circ}-\sin{23^\circ}\sin{37^\circ}=\frac{1}{2}\)
    2. \(\tan{(\frac{\pi}{4}+x)}\times \tan{(\frac{\pi}{4}-x)}=1\)
    3. \(2\cos{(\frac{3\pi}{2}-x)}\times \cos{(\pi+x)}=\sin{2x}\)

  8. Prove the following:

    1. \(\sin^2{5x}-\sin^2{3x}=\sin{8x}\sin{2x}\)
    2. \(\frac{\sin{x}+\sin{2x}+\sin{3x}}{\cos{x}+\cos{2x}+\cos{3x}}=\tan{2x}\)
    3. \((\cos{3x}+\cos{x})^2+(\sin{3x}+\sin{x})^2=4\cos^2{x}\)

  9. If \(\tan{x}=2\), then find \(\tan{2x}\), \(\sin{2x}\), and \(\cos{2x}\).

  10. Find \(\sin{\frac{x}{2}}\), \(\cos{\frac{x}{2}}\), \(\tan{\frac{x}{2}}\), if \(cos{x}=\frac{7}{25}\).