4.3 Equivalent trigonometric Expressions

Today, in class we learned about equivalent trigonometric expressions. Now you all know that sin(π/6)= 0.5 and in today's lesson you will learn how cos(π/3) also equals 0.5. This means that the x-value, (π/6) radians in regards to the sine graph and the x-value, (π/3) radians in regards to the cosine graph both create the same y-value. I found this lesson very interesting and in this section you will come across many trigonometric identities, which hopefully you have learned in grade 11 functions. These trigonometric identities will help you understand how the above equations are equal and you will apply your understanding of these trigonometric identities in questions asking for you to use equivalent trigonometric expressions. In grade 12, there are many different types of trigonometric identities and Ms. Burchat will give you a sheet, which will show you the trigonometric identities that you will be using as a reference throughout this unit. The identities mostly used in this section are Co function identities for angles in the 1st and 2nd Quadrant. This means that you will use a specific trigonometric identity for the angle found in either the 1st or 2nd quadrant. An example done in class of how we get to some of these identities in both categories, is written below:
1st. Quadrant:
1. sin30º= cos60º, both approx. = 0.5000
2. sin50º= cos40º, both approx. = 0.7660

2nd. Quadrant:
1. -sin20º=cos110º, both approx. = -0.3420
2. sin100º=cos10º, both approx. = -0.1736
Key: degrees is substituted for the x value

We have now come up with the follwing identities:
1st Quadrant:
1. sinx = cos((π/2)-x)
2. cosx = sin((π/2)-x)

2nd Quadrant:
1. -sinx= cos(x+(π/2))
2. cosx= sin(x+(π/2))

key: subtract x from (π/2) and keep answer in radians
These identities connect the cosine and sine of an angle. This means that determining the x value in the cosine part of the equation will help you determine the x-value of sine. This method is used throughout the whole section of lesson 4.3.

Also, keep in mind this important aspect in order to get a better understanding of how these co-function identities in the 1st and 2nd quadrant are created.

cosx= sin(x+(π/2)): This means that doing a phase shift to the left (π/2) radians/units to the sine(x) function, creates the cos(x) function. Always think about transformations when dealing with all the other Co function identities as well. It happens for both the Co function identities for angles in the first quadrant and the Co function identities for angles in the second quadrant.