Chapter 2: Trigonometry
2.1 Angles in Standard Position
We often think of angles as being made when two lines originate from the same point. So as to investigate angles further, there exists a need for a standard upon which we can base our work.
Definitions:
Initial arm -
Terminal arm -
Angle in standard position -
The coordinate axes conveniently provide 4 quadrants, and they can each hold a particular range of angles:
Many trigonometric properties have a basis in the range of to . The Cartesian plane, however, has a range of to . We therefore declare there to be a reference angle, between and , related to a given angle.
Definition:
Reference angle -
The reference angle is denoted and can be found using a formulaic approach in each quadrant.
Quadrant 1:Quadrant 2:
Quadrant 3:Quadrant 4:
Example: State the reference angle for each provided angle.
Special Right Triangles
There exist two right triangles which we can use to determine the exact values of the trigonometric ratios (sine, cosine, and tangent) for the angles , , and .
From these triangles, we can state:
That’s all for now.
Homework: textbook pages 83-87, #1-6, 10, 22
Chapter 2: Trigonometry
2.2 Trigonometric Ratios of Any Angle
Let be any angle in standard position in quadrant I. We will also let be a point on the terminal arm of .
We can say, that for this angle in QI:
We can still use those definitions for sine, cosine, and tangent in the other three quadrants. This leads us to the development of the handy-dandy CAST rule:
Example: The point (5, -12) lies on the terminal arm of . Determine the exact trigonometric ratios for sin, cos, and tan .
If we are tasked with finding the value of a trigonometric ratio for a particular angle, we first look at that angle’s reference angle, find the ratio for that, and then declare it as positive or negative based on its quadrant.
Example: Find the exact value of
Example: Find the exact value of .
There are cases where we have to extract information to determine which quadrant lies in.
Example: Given and determine the values of sin and tan .
If we are to find the angle itself, rather than the associated trigonometric ratios, we should:
- First determine the quadrant(s) where the solution lies
- Solve for the reference angle using the positive version of the given ratio
- Use the reference angle to determine the final answer
Example: Solve for , given and .
Example: Solve for , given .
Example: Solve for , given that P(-4, 6) lies on the terminal arm of .
Homework: textbook pages 96-99, #2-7, 9, 11, 14, 18, 22, 29
Chapter 2: Trigonometry
2.3 The Sine Law (Part 1)
Whenever we scale a triangle, the angles always remain the same. That means that the trigonometric ratios for those angles remain the same, and thus there exists a constant proportionality (for that triangle) between its sides and its angles.
Let us look at one such ratio:
Definition:
Sine law -
We can use define this for other triangles as well.
For DEF:For XYZ:
Example:
Solve for , to the nearest tenth.
Example:
Solve for to the nearest tenth.
Example:
Solve for to the nearest tenth of a degree.
Homework: textbook pages 108-109, #1-4, 12, 13
Chapter 2: Trigonometry
2.3 The Sine Law (Part 2)
Suppose we are to draw triangle ABC with parameters . There are two possibilities!
What we have here is an ambiguous case. The triangle’s exact shape hasn’t been given to us; we’ve just been given information in the form angle-side-side. That is, we have an angle, a side connected to that angle, and a detached side. No information is given about the other angles, and that opens the door to having two possible triangles. First, however, we need to determine if that is, in fact, the case we are working with.
If is acute, then:
- If , then we have no solution
- If , then we have one solution (a right triangle)
- If , we have one solution
- If , we have two solutions
If is obtuse, then we only have a solution if , and no solution otherwise.
It’s a ton of fun.
Example: Solve the triangle ABC where , , and .
Example: Solve the triangle XYZ where , , and .
Homework: textbook pages 108-112, #5, 6, 8-11, 20, 24
Chapter 2: Trigonometry
2.4 The Cosine Law
The sine law assists us in the situation where, when we have two angle/side pairs, there is only one unknown value. There exist situations where the sine law may not be able to help us directly, and we thus need another weapon in our mathematical arsenal so that we never find ourselves stymied.
Check out these triangles:
Every Batman needs a Robin. That’s where the cosine law comes into play. We develop it as follows:
This version of the cosine law is of use to us when we are solving for the side opposite the one angle we are given. If we are given all three sides and need to find an angle, we can manipulate the formula as follows:
Example: In triangle ABC, , , and . Solve for side , to the nearest tenth.
Example: In triangle XYZ, , , and . Find the measure of , to the nearest tenth of a degree.
If you have to completely solve a triangle and must initially use the cosine law, you can note that once it has been used, you have multiple options for solving the rest of the triangle.
Homework: textbook pages 119-124, #1-2 (a, c only), 3, 4 (a, c, e only), 5, 10, 21, 24, 31
Textbook review: pages 126-128, #1-7, 9-12, 16, 19-23