This week we went over a plethora of things such as angles in standard position in all the quadrants, co-terminal angles, and exact trig ratios and special triangles. The majority of the stuff we learnt this week were new and concepts that I wasn’t very familiar with because they weren’t in the Math 10 curriculum except Pythagoras Theorem and SOH CAH TOA. Although, there were some concepts that were a build-on from past years in elementary and middle school.
For our first lesson, we went over how to determine the trigonometric ratios of an angle given a terminal point:
Here is an example of a problem:
The point P (4, 7) is on the terminal arm of an angle θ in standard position. Also, standard position just means one ray is on the positive x-axis and it’s also measured counterclockwise.
a) Determine the distance r from the origin to P
b) Determine the primary trigonometric ratios of θ
c) Determine the measure of θ to the nearest degree
First, you draw a diagram representing all of the information. This point is located in quadrant one because both the x and y value are positive. You also label where the point P is located, plus the length of the sides of the triangle. The hypotenuse is the side length that is missing its value, so it can just be represnted by an “r,” since it’s also the value of the radius. As, you could draw a circle from the terminal point and all the way back to it.
Example:
In addition, the degree of the angle can be represented by the theta symbol.
Also, the ray from O to P is the terminal arm of the angle and the point P is the terminal point for that angle.
This is how the diagram with the labels should look like:
a) For part A, to find the length of r (the distance from the origin to P), you have to use Pythagorean Theorem, as the triangle drawn is a right triangle.
b) Find the primary trigonometric ratios of the angle:
Label the triangle to determine which sides are the hypotenuse, adjacent, and opposite:
So, the hypotenuse is the longest side of the triangle, the opposite is the side that is opposite to the angle, and the adjacent side is the remaining side.
Sin θ = opposite over hypotenuse or y/r
Sin θ = 7/√65
Cos θ = adjacent over hypotenuse or x/r
Cos θ = 4/√65
Tan θ = opposite over adjacent or y/x
Tan θ = 7/4 or 1.75
c) To determine the measure of θ to the nearest degree, you can use any trigonometric ratio from part b and it will give you the answer.
The measure of θ is 60 degrees approximately.
Here is another example of determining the trigonometric ratios of an angle given a terminal point:
The point P (3, 4) is on the terminal arm of an angle in standard position.
a) Determine the distance r from the origin to P
b) Determine the primary trigonometric ratios of θ
c) Determine the measure of θ to the nearest degree
First, you draw a diagram again to visualize a triangle with the point (3, 4) which is also located in quadrant one, since both the x and y values are positive.
Here is how the drawing should look like:
a) Determining the distance from the origin to P
You have to once again use the Pythagorean Theorem to find the length of r.
This time the value of r is a whole number, since 25 is a perfect square.
b) The primary trigonometric ratios of θ
Label the triangle to determine the hypotenuse, adjacent, and opposite side:
Sin θ = opposite over hypotenuse
Sin θ = 4/5
Cos θ = adjacent over hypotenuse
Cos θ = 3/5
Tan θ = opposite over adjacent
Tan θ = 4/3
c) Determine the measure of θ to the nearest degree
Use any trigonometric ratio from part b to determine the angle to the nearest degree:
The measure of θ is 53 degrees approximately.