and how to label right triangles:

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1 Grade 9 IGCSE A1: Chapter 6 Trigonometry Items you need at some point in the unit of study: Graph Paper Exercise 2&3: Solving Right Triangles using Trigonometry Trigonometry is a branch of mathematics that investigates the relationships between the sides and angles of triangles. Right Triangle Trigonometry focuses on the relationships between the ratio of two sides and non-right angles of RIGHT triangles. Let s look at the names of the parts of the triangle: and how to label right triangles: What do Capital Letters refer to? What do lower case letters refer to? Notice the placement of the letters for the lengths relative to the letters for the angles. Greek letters are often used to name angles theta (θ) and alpha (α) or beta (β) The sides (Opposite, Adjacent) are named relative to the non-right angle that is selected. By definition: opp adj opp sin a = ; cosa = ; tana = these are read: hyp hyp adj sine alpha; cosine alpha; tan alpha Do you know the mnemonic SOHCAHTOA? A trig function takes an angle and provides a ratio of the two established sides of the triangle. As a result, there is no such thing as a trig function without an angle eg sin In any trig equation, there are three parts of a triangle that are related one angle and two sides. Given two of the three pieces of information, you can determine the value all of the angles and sides of the right triangle. Use your calculator to help you. Remember that all prior knowledge still applies: Geometry Theorems: AAAAAA = 180 Pythagorean Theorem, etc. In Exercise 3, all values must be exact until the last step. What is the difference between these three? 10ccc45 ; ccc45 10; cos (45 10) We will take a look at Exercise 3 #2. Exercise 2&3: C-type: Exercise 2 16, 19, 23, 24 Exercise 3 3, 6, 9 A-type: Exercise 3 12

2 Exercise 4: Finding an angle of a Right Triangle given two sides By definition: adj opp opp cos a = ; sina = ; tana = hyp hyp adj Trig functions: angle ratio Inverse Trig functions: ratio angle Then the inverse trig functions are defined as: = adj 1 1 opp opp a cos ; a = sin ; a = tan 1 hyp hyp adj Given two sides of a triangle, you can determine the angle make sure that you calculator is set to degrees and NOT radians. Exercise 4: C-type: 3, 6, 16, 18 A-type: 24, 25 Bring Graph Paper and Protractor to the next class.

3 Exercise 5: Bearings and Angles of Elevation and Depression Bearing is a method to determine direction measured clockwise from North. A bearing is always written with 3 non-decimal digits. For example, 60 and 060, while both have a measure of 60 degrees, the starting direction is different and this is essential to recognize. BTW, what is the starting direction of 60? For every segment of a journey, you can assume that the direction and distance is consistent and a change will not occur until stated. At every transition, it is important to include the (NSEW) crosshairs so that you can label angles and also find angles based upon parallel lines as all NS and EW lines will be parallel lines. Then you will need to use Geometry Theorems related to parallel lines to determine other angles to help solve the problem. In words, express the journey shown in the diagram to the right using bearings. Angle of Elevation and Depressions are angles which describe the line of sight from the horizontal. Notice that one result is the diagram to the right. What relationship exists between the angle of depression and the angle of elevation? In these problems, it is appropriate to round each calculation to 3 s.f. Exercise 5: C-type: 3, 6, 8, 12, 13, 16 A-type: 24, 25 Bring Graph Paper to the next class.

4 Exercise 6&7: Scale Drawings and 3D problems Be sure to label you diagrams correctly. You must always state the scale. Be sure that you can draw and label 3D diagrams appropriately. Perspective Hidden lines are dotted What clues you in to whether you need to use a 3D approach or a 2D approach? Let s discuss in class. Exercise 6&7: C-type: Exercise 6 2, 4 Exercise 7 3, 4 A-type: Exercise 7 7, 10

5 Exercise 8: The trig value of any angle Initially, we limited ourselves to the study of Right Triangle Trigonometry. Now we want to extend our definitions to include any angle. The UNIT CIRCLE is the basis of all trig functions. The Unit Circle is again defined as a circle with r = 1 with center at the origin and satisfies the equation (prove this to be true). The equation of a circle with radius r and center at (0,0) is. Angle in standard position has The initial side on the positive x-axis. Vertex at the origin. Terminal side Angles are considered positive if the rotation from the initial side is counter-clockwise; negative if clockwise. As any point P(x, y) moves on the unit circle, The x-coordinate of P is cos The y-coordinate of P is sin Provided that is the angle made by OP with the positive x-axis. The end result are these two graphs for y = ssnx aaa y = cccc take a look at why this is true: Demo Exercise 8: C-type: 1, 2, 3 9 all, 11, 12 A-type: 14

6 Exercise 9-11: The Sine and Cosine Rule If you wanted to find the length of a side or the value of an angle in the right triangle, we would just use right triangle trigonometry. But if the triangle was not a right triangle, then we would use either the Sine Rule or Cosine Rule. Sine Rule: Given any triangle, the Sine Rule finds the relationship between any two sides and the angles opposite the sides in the triangle. IS: Notice again the notation used for angles and sides. How do you prove the Sine Rule is true given any AAA? Cosine Rule: The cosine rule allows you to determine the length or angle in a non-right triangle given two sides and the included angle. Given any triangle, the cosine rule states: IS: Use the same triangle above to prove the Cosine Rule given any AAA? The Cosine Rule relates three sides of the triangle and one angle. Therefore, if you are given 3 of the 4 pieces of information, you can always find the forth. Exercise 9&10: C-type: Exercise 9 7, 9, 19, 21 Exercise 10 7, 9, 19, 21 A-type: Exercise 11 5, 8, 12 Chapter 7 is about graphs, so be sure to have graph paper for all the exercises for the chapter.