Exercise Set 4

These exercises cover the topic of Trigonometry.

Tip: always start by drawing a labelled diagram in trigonometry questions.

  1. Consider the smaller of the two angles between the hour hand and minute hand of a clock (remembering that the hour hand moves continuously!). Write the angles at the following times in both degrees and radians (in terms of \(\pi\)).

    1. 6:00
    2. 3:00
    3. 4:00
    4. 4:30
    5. 6:45

  2. Convert the following angles from degrees to radians. Give answers both in terms of \(\pi\) and to 2 d.p.

    1. \(330^\circ\)
    2. \(22.5^\circ\)
    3. \(27^\circ\)
    4. \(35^\circ\)

  3. A bearing is the angle measured clockwise from North to the direction of interest (note how this differs to how we measure angles in the coordinate plane). A point \(K\) is 12km due west of a second point \(L\) and 25km due south of a third point \(M\). Calculate the bearing of \(L\) from \(M\) (in degrees).

  4. Solve (i.e. find all unknown angles and side lengths) the triangle \(ABC\) where \(A = 53^\circ\), \(B = 61^\circ\) and \(a = 12.6\)cm.

  5. Solve (i.e. find all unknown angles and side lengths) the triangle \(ABC\) where \(a=10.2\)m, \(c=14.6\)m and \(C = 32.5^\circ\).

  6. Let \(AOB\) be a triangle. \(OA = 60\)mm, \(AB = 180\)mm and \(OB = 200\)mm. Find angle \(A\).

  7. An angle of elevation is an angle that an imaginary straight line must be raised from the horizontal ground to line up with a point of interest above the ground. An observer is standing at a point \(M\) which is \(30\)m from the base of a tower. On top of the tower is a vertical mast. If the angles of elevation of the top of the tower and the top of the mast from \(M\) are \(40^\circ\) and \(50^\circ\) respectively, calculate the height of the mast.

  8. The small hand of a clock is 75% the length of the long hand, which has length \(x\). Calculate the distance between the ends of the hands at 5 o’clock (in terms of \(x\)).

  9. A student \(1.8\)m tall is standing \(24\)m away from a tree and using a eye level instrument to measure the angle of elevation. The angle measured to the top of the tree is \(12^\circ 34'\), calculate the height of the tree. (Degrees can be further subdivided in to minutes denoted \(x'\) and seconds denoted \(x''\), with \(1'\) being \(1/60\) of a degree and \(1''\) being \(1/60\) of a minute. To use a calculator you will first need to convert minutes and seconds to decimals.)

  10. The angles of elevation of a navigation balloon that is flying in between two points on the ground \(A\) and \(B\) are \(48^\circ\) and \(62^\circ\) respectively. If \(A\) and \(B\) are \(0.3\)km apart, calculate the height of the balloon.

  11. The figure below shows a tetrahedron with an equilateral triangle of side 2m forming the base and isosceles triangles of equal side 3m forming the slanting faces. Calculate:

    1. The height of the tetrahedron \(ND\);
    2. The angle that edge \(DA\) makes with the plane \(ABC\);
    3. The angle between the planes \(ACD\) and \(ACB\).

  1. This is a harder question using trigonometric identities. Here we shall derive a special (slightly simpler) case of the sum of two sinusoidal waves of the same frequency that was shown in lectures. We will consider two waves of the form \[y_1=A_1\sin(\omega t),\qquad y_2=A_2\cos(\omega t)\].
    1. First assume we can write \(y_1+y_2\) in the form \(A\sin(\omega t+\phi)\). Then using the trigonometric identity \[\sin(\theta+\phi)=\sin(\theta)\cos(\phi)+\cos(\theta)\sin(\phi)\] deduce that \(A_1=A\cos(\phi)\) and \(A_2=A\sin(\phi)\).
    2. Now using the trigonometric identity \(\sin^2(\theta)+\cos^2(\theta)=1\) show that \(A^2=A_1^2+A_2^2\).
    3. Finally, using \(\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}\), show that \(\tan(\phi)=\frac{A_2}{A_1}\).
  2. Using the formulas derived in the previous question for \(A\) and \(\phi\), write the following in the form \(A\sin(\omega t + \phi)\).
    1. \(-2\sin(\omega t) + 5\cos(\omega t)\)
    2. \(-5\cos(\omega t) + 5 \sin(\omega t)\)
  3. In a spring-mass system the motion of the mass is described by \[x=A_1\cos(\omega t)+A_2\sin(\omega t)\] where \(x\) is the distance of the mass from its equilibrium position, \(\omega\) is the natural frequency of oscillations, and \(A_1\) and \(A_2\) are constants. For \(A_1=1\), \(A_2=\sqrt{3}\) and \(\omega=10\,\text{rad s}^{-1}\):
    1. Write \(x\) in the form \(A\sin(\omega t+\phi)\) and state the amplitude of \(x\).
    2. Sketch one complete cycle of \(x\), labelling \(A\), \(\phi\) and the period \(T\).