Trigonometry Angles of Elevation and depression 1) If the angle of elevation of the top of a vertical 30m high aerial is 32, how is it to the aerial? 2) From the top of a vertical cliff 80m high the angles of depression of two buoys lying in front of the cliff face are 23 and 15, respectively. How far are the buoys apart? 3) From a ship at sea the angles of elevation of the top and bottom of a vertical lighthouse standing on the edge of a vertical cliff 31 and 26, respectively. If the lighthouse is 25m high, find the height of the cliff. 4) From a window 4.2m above horizontal ground the angle of depression of the foot of a building across the road is 24 and the angle of elevation of the top of the building is 34. Determine the width of the road and the height of the building. 5) The elevation of a tower from two points in front of the tower are 20 and 24, respectively and the points of observation are 300m apart, find the height of the tower. Sine and Cosine rule Find all the remaining sides, angles and area of the following triangles 6) A = 29, B = 68, b = 27mm 7) B = 71, C = 56, b = 8.6cm 8) d = 17cm, f = 22cm, F = 26 9) d = 32.6mm, e = 25.4mm, D = 104 10) j = 3.85cm, k = 3.23cm, K = 36 11) k = 46mm, l = 36mm, L = 35 12) q = 12cm, r = 16cm, P = 54 13) q = 3.25m, r = 4.42m, P = 105 14) x = 10cm, y = 8cm, z = 7cm 15) x = 21mm, y = 34mm, z = 42mm 16) Two sides of a triangular plot of land are 52m and 34m respectively. If the area of the plot is 620m 2 find a. The length of fencing needed to enclose the plot b. The angles of the triangular plot 17) A building site is in the form of a quadrilateral as shown below. The total area is 1510m 2. Determine the length of the unknown side and the perimeter of the building site.
18) Determine the length of members BF and EB in the roof truss shown below 19) PQ and QR are the phasors representing the alternating currents in two branches of a circuit. Phasor PQ is 20A and is horizontal. Phasor QR (which is joined to the end of PQ to form triangle PQR) is 14A and is at an angle of 35 to the horizontal. Determine the resultant phasor PR and the angle it makes with PQ. Trigonometric waveforms CAST rule 19) Convert the following angles to radians a. 30 b. 45 c. 60 d. 55 e. 210 f. 150 g. 330 20) Convert the following angles from radians to degrees a. 2.45
b. 0.34 c. 1.57 d. 2.78 e. 5.67 21) Determine all the angles between 0 and 360 whose Sine is a. 0.6792 b. -0.1483 c. 0.866 d. -0.9397 22) Solve the following equations for values of x between 0 and 360 a. x = Cos -1 0.8739 b. x = Cos -1 (-0.5572) c. x = Cos -1 (0.5) d. x = Cos -1 (-0.8) 23) Find the angles between 0 and 360 whose tangent is a. 0.9278 b. -2.3418 c. 1.428 d. -1.732 24) State the amplitude and periodic time of the waveform and sketch the curve between 0 and 360 a. y = Cos3A b. 5x y = Sin 2 c. y = Sin4t d. y = 3Cos 2 e. 7 3x y = Sin 2 8 f. o y = 6 Sin( 45 ) g. y = 4 Cos(2θ + 30 o ) 25) In this problem find the amplitude, periodic time, frequency and phase angle stating whether it is leading or lagging Sinwt of the alternating quantities. a. i = 40 Sin(50π t+ 0.29) ma b. y = 75 Sin(400.54) cm c. v = 300 Sin(200π 0.412)V d. v = 90 Sin(400 πt) V
e. i = Sin(100π t+ 0.3) A f. e= 200 Sin(628.40.41) V 26) Sketch the resulting waveform for each of the waveforms represented in problem 23 27) A sinusoidal voltage has a maximum value of 120V and a frequency of 50Hz. At time t =, the voltage is a. Zero b. 50V Express the instantaneous voltage v in the form v = ASin( wt± α ) 28) An alternating current has periodic time of 25ms with a maximum value of 20A. When time t = 0, current i = -10A. Express the current i in the form i = ASin( wt± α ). 29) An oscillating mechanism has a maximum displacement of 3.2m and a frequency of 50Hz. At time t = 0 the displacement is 150cm. Express the displacement in the general form ASin( wt ± α ). 30) The current i in an ac circuit at any time t seconds is given by: i = 5 Sin(100π 0.432) A Determine a. The amplitude, periodic time, frequency and phase angle (in degrees) b. The value of the current when t = 0 c. The value of current at t = 8ms d. The time when the current is first at a maximum e. The time when the current first reaches 3A f. Sketch one cycle of the waveform showing relevant points 31) The voltage v in an ac circuit at any time t seconds is given by: v = 3.7 Sin(50π 0.24) V Determine g. The amplitude, periodic time, frequency and phase angle (in degrees) h. The value of the voltage when t = 0 i. The value of voltage at t = 2.5ms j. The time when the voltage is first at a maximum k. The time when the voltage first reaches 1.4A l. Sketch one cycle of the waveform showing relevant points
32) The instantaneous voltage of an ac. circuit at any time t is given by v = 100 Sin(50π 0.523) V. Find a. The peak to peak voltage, the periodic time, the frequency and phase angle in radians. b. The voltage when t = 0 c. The voltage when t = 8ms d. The times in the first cycle when the voltage is 60V e. The times in the first cycle when the voltage is -40V f. The first time when the voltage is maximum 33) An alternating current varies with time over half a cycle as follows Current (A) 0 0.7 2.0 4.2 8.4 8.2 2.5 1.0 0.4 0.2 0 Time (ms) 0 1 2 3 4 5 6 7 8 9 10 The negative half cycle is similar. Plot the curve and determine a. The frequency b. The instantaneous values at 3.4ms and 5.8ms 34) The instantaneous vales of two alternating voltages are given by v1 = 5 Sin( wt) π and v2 = 8Sin w 6. By plotting v 1 and v 2 on the same axes, using the same scale over one cycle, obtain the waveform by the addition of v 1 and v 2.