Worksheet 3.3. Trigonometry. Section 1. Review of Trig Ratios

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1 Worksheet. Trigonometry Section Review of Trig Ratios Worksheet. introduces the trig ratios of sine, cosine, and tangent. To review the ratios, consider a triangle ABC with angle φ as marked. B c a φ A b C The hypotenuse (hyp) of the triangle is c; the adjacent (adj) side is b; the opposite (opp) side is a. The side of length a is opposite the angle, and the side of length b is the side adjacent to the angle which is not the hypotenuse. Then we have sin φ = opp hyp = a c cos φ = adj hyp = b c tan φ = opp adj = a b Note also that a sin φ cos φ = c b c = a b = tan φ Eercises:. For the following triangle, find the ratios: (a) (b) (c) sin tan cos 5 4

2 . For the following triangle, find the ratios: (a) tan (d) sin φ (b) cos φ (e) tan φ (c) sin (f) cos a c b φ. (a) Use Pythagoras theorem to find (b) Find (i) sin (ii) tan (iii) cos 5 Section Degrees and Radians Recall from Worksheet.9 that = π 0 radians In university maths it is much more common to give angles in radians rather than degrees. If the units are left off an angle, then the angle is in radians. Degrees can be converted to radians using the above formula, but it will be very convenient for you to know some standard conversions. In particular: 90 = π 0 = π 45 = π 4 0 = π 0 = π 0 = π Eample : An equilateral has three equal angles of π. Eample : Convert 50 to radians. 50 = 50 π 0 = 5π radians Eample : How many degrees is π 9 radians? We know 0 = π, so π 9 = 0 9 therefore π 9 = 0

3 If we think of an angle as the amount of rotation of a straight line about the angle, then we can define a positive rotation as one which is anti-clockwise and a negative rotation as one which is in a clockwise direction. We can see the ordinary y plane with the verte of the angle at the origin and the base of the angle beginning at the positive -ais. i.e. the positive -ais represents the position of the line if the angle of rotation is 0. So for the angle we get the following diagram: y We can now represent angles graphically and we can deal with angles of any size. Recall that a full revolution is π, or 0. So rotating a line through π will bring it back to its starting position. Eample 4 : Represent π radians graphically. Since π is the angle half way round the plane, π is the angle one third of the way around the upper half of the plane. y π Eample 5 : Represent π radians graphically. y π Eample : Represent π 4 radians graphically. y π 4

4 Eercises:. Write the following degrees in radian measure (a) 0 (b) 0 (c) 90 (d) 70 (e) 5. Convert the following radian measures to degrees (a) π (b) π 9 (c) π 4 (d) 5π (e) 7π Section Standard Triangles There are two triangles which are known as standard triangles. These triangles and the information contained in them should be memorized, as you will be epected to know certain information without using a calculator. The first triangle is a right-angled isosceles triangle. Recall that an isosceles triangle has two angles the same and two sides the same length: π 4 The associated trig ratios are fairly simple to work out, and are left as eercises. The second standard triangle is half an equilateral triangle of length. π π When we have taken half the equilateral triangle, we end up with the following: π π π Pythagoras theorem gives us the length of the vertical side as, and the angle φ is half the top angle so φ = π. The trig ratios given by this triangle are: 4

5 sin π = sin π = cos π = cos π = tan π = tan π = Once the triangles are memorized the trig ratios can be found by just drawing either of the two triangles. It is important that you memorize the trig ratios for the angles π, π and π 4. Eercises:. Find the eact ratios for the following (a) tan π (b) cos π (c) sin π 4 (d) tan π 4 (e) tan π. Use eact ratios to find in each of the following equations, where 0 π. (a) sin = (b) cos = (c) tan = (d) sin = (e) cos = Section 4 Using trigonometric ratios We can use the trigonometric ratios described in the previous sections to find an unknown angle or side in a right-angled triangle. Consider the following triangle: Let us say that we wish to find in this triangle. The side that is units long is adjacent to ; the side that is units long is opposite to, so we have tan = opposite adjacent = = 4 5

6 Consequently, is an angle whose tangent is 4. That is = tan 4. By using the tan button on a calculator, we find that = 0.97, to three decimal places. Note that this answer is in radians. Eample : What is? 0 7 With respect to, the opposite side is 7 units long, and the hypotenuse is 0 units long. Therefore, sin = opposite hypotenuse = 7 0. = 0.5 The last step was carried out using the sin button on a calculator, and the answer is approimate and in radians. Eample : What is? 0 0. The trigonometric ratios may also be used to find the length of a side in a right angled triangle. sin 0. = OPP HYP = 0 = 0 sin 0. = 5. Eample : What is? 5 0.4

7 tan 0.4 = OPP ADJ = 5 = 5 tan 0.4 = 0.7 Eercises:. Find in each of the following 4 (a) (b) 5 5 (c) (d).4.7. Find, to decimal places, in each of the following triangles π (a) (b) (c) (d) π 5.4 Section 5 Inverse Trig Functions Sometimes you will come across the notation sin a or cos a. Now, sin a does not mean. It is called the arcsine of a, and means this: the sin of what angle will give an answer a? sin a So sin a = means sin = a The same rule applies to cos a and tan a. If you wish to write as (sin a) so that there is no confusion. sin a then you would do so 7

8 Eercise. Trigonometry. (a) Covert i. 5π to degrees ii. π 9 to degrees iii. 0 to radians; write the answer as a number times π. iv. 4 to radians; write the answer to decimal places (b) Find the eact values of i. sin π 4 ii. cos π. iii. tan π (a) Find the value of (b) Evaluate. Joan walks 5km north, then.km east. (a) Put these distances onto the appropriate sides of the triangle below: (b) Find the angle, the bearing that Joan has effectively walked along.

9 Answers.. (a) i. 50 ii. 40 iii. 4π 9 iv. 7π 0 = 0.7 (b) i. ii. iii.. (a) 0 (b) π. (a) 5 km. km (b)