08/01/2017. Trig Functions Learning Outcomes. Use Trig Functions (RAT) Use Trig Functions (Right-Angled Triangles)

Transcription

1 1 Trig Functions Learning Outcomes Solve problems about trig functions in right-angled triangles. Solve problems using Pythagoras theorem. Solve problems about trig functions in all quadrants of a unit circle. Solve problems about trig functions of angles greater than 360 o and less than 0 o. 2 Use Trig Functions (Right-Angled Triangles) Recall in a right-angled triangle, we name three sides based on the angle of interest: Hypotenuse is always the longest side. Opposite is literally opposite the angle of interest. Adjacent is literally adjacent to the angle of interest. 3 For a particular angle, the ratio between the side lengths is the same for every triangle, regardless of size. The ratio of the opposite to the hypotenuse is called sine: sin A = opposite hypotenuse The ratio of the adjacent to the hypotenuse is called cosine: cos A = adjacent hypotenuse The ratio of the opposite to the adjacent is called tangent: tan A = opposite adjacent 1

2 4 These are given on page 13 of the Formula and Tables Book, but are presented without the words opposite, adjacent, or hypotenuse. 5 e.g. Calculate sin(37), cos(37), and tan(37) using the triangle below: 6 e.g. Use a calculator to calculate each of the following and fill in the blanks. 1. sin 60 = 0.866, so the opposite is times as long as the hypotenuse. 2. cos 28 =, so the is times as long as the. 3. tan 76 =, so the is times as long as the. 4. sin 62 =, so the is times as long as the. 2

3 7 1. Draw a triangle ABC so that AC = 100, ABC = 90 o, and CAB = 50 o. Find AB and BC. 2. Draw a triangle DEF so that DE = 7, DEF = 90 o, and FDE = 25 o. Find DF and EF. 3. Draw a triangle LMN so that LM = 62, LMN = 37 o, and MNL = 90 o. Find LN and MN. 8 While the trig functions apply to angles and return the ratio of side lengths, there are inverse trig functions that apply to ratios and return angles. If sin(a) = opposite opposite, then A = sin 1 hypotenuse hypotenuse If cos A = adjacent adjacent, then A = cos 1 hypotenuse hypotenuse If tan A = opposite, then A = tan 1 opposite adjacent adjacent On a calculator, these are typed using e.g. SHIFT+sin sin 1 B is pronounced inverse sine of B or sine minus one of B, likewise for cos 1 B and tan 1 B. 9 e.g. Given the triangle shown, find θ: tan(θ) = θ = tan θ 26.6 o 3

4 10 1. Draw a triangle ABC so that AB = 9, BC = 4, and CBA = 90 o. Find BAC. 2. Draw a triangle DEF so that DF = 13, EF = 7.5, and DEF = 90 o. Find FDE. 3. Draw a triangle LMN so that LN = 2.2, LM = 1.7, and LMN = 90 o. Find NLM. 11 In real world problems, two addition terms are important. When looking up, the angle from the horizontal is called the angle of elevation. When looking down, the angle from the horizontal is called the angle of depression. Additionally, a clinometer is a device used to measure these angles From the top of a light house 60 meters high with its base at the sea level, the angle of depression of a boat is 15 degrees. What is the distance of boat from the foot of the light house? 2. The angle of elevation of the top of an incomplete vertical pillar at a horizontal distance of 100 m from its base is 45 degrees. If the angle of elevation of the top of the complete pillar at the same point is to be 60 degrees, then the height of the incomplete pillar is to be increased by how much? 3. A 10 meter long ladder rests against a vertical wall so that the distance between the foot of the ladder and the wall is 2 meter. Find the angle the ladder makes with the wall and height above the ground at which the upper end of the ladder touches the wall. 4

5 13 Use Pythagoras Theorem Pythagoras Theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. pg Use Pythagoras Theorem Find the missing side length in each of the following triangles pg The right-angled triangle definitions of sine, cosine, and tangent limit the possible angles between 0 and 90 degrees. To generalise the trig functions, we redefine them based on the unit circle. The unit circle is a circle on the coordinate plane with centre (0, 0) and radius 1. 5

6 16 Measuring the angle anti-clockwise from the positive x-axis: cos A is the x-coordinate where the radius meets the circumference, sin A is the y-coordinate where the radius meets the circumference. 17 Using the unit circle, complete this table. A (degrees) sin A cos A tan A Recall that angles measure anti-clockwise from the positive x-axis. cos A is the x-coordinate sin A is the y-coordinate sin A tan A = cos A 18 Draw a unit circle including axes. Draw radiuses at 60 o, 120 o, 240 o, and 300 o. Find cos A for each of the angles drawn. Find sin A for each of the angles drawn. 6

7 19 Each angle can be reframed as the smallest angle made with the x- axis, called the reference angle. For 60 o, 120 o, 240 o, and 300 o, this angle is 60 o. Since cos 60 = 0.5, cosine of each of the other angles is either 0.5 or -0.5 depending on what side of the circle it s on. 20 Recall that cos A represents the x- coordinate of the circumference. Thus, it is positive in the first and fourth quadrants of the circle. Similarly, it is negative in the second and third quadrants of the circle. 21 Similarly, sin 60 = 0.866, so the sine of each of the other angles is either or

8 22 Recall that sin A represents the y- coordinate of the circumference. Thus, it is positive in the first and second quadrants of the circle. Similarly, it is negative in the third and fourth quadrants of the circle. 23 Recall that tan A sin A represents. cos A Thus, it is positive in the first and third quadrants of the circle Similarly, it is negative in the second and fourth quadrants of the circle. (You can think of it as the slope of the radius). 24 If cos A = 1 2, find two values of A if 0o A 360 o. If sinb = 1 2, find two values of B if 0o B 360 o. If tan C = 3, find two values of C if 0 o C 360 o. 8

9 25 Solve > 360 o and < 0 o Problems Sine, cosine, and tangent are periodic. Each function repeats itself every 360 o. e.g. cos 400 = cos( ) = cos 140. In general, cos A = cos A + 360n sin A = sin(a + 360n) tan A = tan(a + 360n) pg Solve > 360 o and < 0 o Problems Likewise, if we measure negative angles (going clockwise instead of anticlockwise), the result is equivalent to some anticlockwise angle. In general, cos A = cos A + 360n sin A = sin(a + 360n) tan A = tan(a + 360n) with negative n pg Solve > 360 o and < 0 o Problems Find: tan 495 o sin 840 o cos( 120 o ) pg

10 28 Solve > 360 o and < 0 o Problems e.g. If cos 3A = 0.5, find all values of A if 0 o A 360 o. cos = 60 o, so our reference angle is 60 o. Cosine is positive in the first and fourth quadrants, so: n 3A = n 3A = n 0 3A = 60 3A = 300 A = 20 o A = 100 o 1 3A = A = 420 A = 140 o 2 3A = A = 780 A = 260 o 3A = A = 660 A = 220 o 3A = A = 1020 A = 340 o pg Solve > 360 o and < 0 o Problems If cos 2A = 1, find two values of A if 2 0o A 360 o. If sin3b = 1, find two values of B if 2 0o B 360 o. If tan 4C = 3, find two values of C if 0 o C 360 o. pg