MHF4U: CPT

Chapter 6

Radians

Radians are what we use as somewhat of an opposite to degrees. We use radians to measure angels just like degrees. Although, radians measure angles based off of the radius of a circle. One radian is the angle formed when the length of the arc of a circle is equal to the radius of the circle. Using radians enables you to express the size of an angle as a real number without any units, often in terms of π.

Real numbers are essentially just numbers that have no decimal places or no long lists of numbers after the decimal. Using real numbers with radians in math helps us preform our equations much easier and with better efficiency and if needed we can just convert our final answer into degrees by the end.

The formula to convert radians into degrees is to multiply your value by by . π/180°

The arc length, a, of a circle is proportional to its radius, r, and the central angle that it subtends, by the formula θ = a/r

Degrees

Degrees are what we usually use to measure angles in a circle. We know, for example, that a whole circle is 360° and half is 180°.

The formula to convert degrees into radians is 180°/π

Special Triangles With Radians and Degrees and on the Cartesian Plane

Special triangle #1:

Hypotenuse: √2
Angle to the right of 90°: π/4 or 45°
Angle above of 90°: π/4 or 45°
Both other sides: 1

Special triangle #2:

Hypotenuse: 2
Side opposite to π/3: √3
Side opposite to π/6: 1
Angle to the right of 90°: π/3 or 60°
Angle above of 90°: π/6 or 30°

The angles in the special triangles can be expressed in both radians degrees. The radian measures can be used to determine the exact values of the trigonometric ratios for multiples of these angles between 0 and 360 degrees. The skills that are used to determine the values of the trigonometric ratios when an angle is expressed in degrees on the Cartesian plane can also be used when the angle is expressed in radians.

Trigonometric ratios are ratios of a triangle that are known as SIN, COS and TAN.

Sin= opposite/hypotenuse
Cos= adjacent/hypotenuse
Tan= opposite/adjacent

Cast rule

• In quadrant 1, All (A) ratios are positive because both x and y are positive.

• In quadrant 2, only Sine (S) is positive, since x is negative and y is positive.

• In quadrant 3, only Tangent (T) is positive because both x and y are negative.

• In quadrant 4, only Cos (C) is positive, since x is positive and y is negative.

Degrees and Radians Wheel

Sin, Cos and Tan Graph

Chapter 7

Exploring Equivalent Trigonometric Functions

As stated by the name topic it covers how different trigonometric functions are equal.

There are many different formulas that are covered in this topic and we use all of them when we are proving identities

Identities Formulas

Compound Angle Formulas:

sin(A±B) = sin(A) cos(B) ± cos(A) sin(B)

cos (A± B)= cos(A)cos(B) ∓ sin(A)sin (B)

cos(A±B)=cos(A)cos(B)∓sin(A)sin(B)

tan (A ± B) = tan (A) ± tan(B)/ 1∓ tan(A) tan(B)

tan(A±B)= 1∓tan(A)tan(B) tan(A)±tan(B)

Used for the following:

- Simplifying expressions involving the sum or difference of angles.

- Solving trigonometric equations where angles are added or subtracted.

- Deriving exact values for non-standard angles

Proving Identities

This section develops techniques for proving trigonometric identities by manipulating one side of the equation until it matches the other. Start with the more complex side of the equation and then use fundamental identities to help solve the puzzle and make the right side equal the left side/ You may also have to take it a step further by factoring expressions or combining fractions.

Solving Quadratic Trigonometric Equations

Substitute your sin(x), cos(x) or tan(x) with a. variable. Solve the quadratic equation using factoring, the quadratic formula, or completing the square.
Substitute back to find x and then use reference angles and periodicity to find all solutions as per the range of what your questions asks.

For example:

If your question is 2sin^2 (x)−sin(x)−1=0, find for π<x<2π, then follow the steps above and use your answer for x to find the reference angles for 2π and π.

Solving Linear Trigonometric Equations

Solving linear trigonometric functions is very similar to solving quadratics. You follow somewhat the same order of finding your x value by isolating your sin x, cos x or tan x and then do the inverse. You then find the values based off of what range the question asks