MHF4U: CPT
Chapter 7
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
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
Proving Identities
Example of Proving (Image #12):
https://docs.google.com/document/d/1ybRVyNDo25rKRhJVBDLO2jNAQiBi7IuvoNgSYJbJbbQ/edit?usp=sharing
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.
Identities Formulas
Ratio Identities Formulas (Image #4): https://docs.google.com/document/d/1ybRVyNDo25rKRhJVBDLO2jNAQiBi7IuvoNgSYJbJbbQ/edit?usp=sharing
Co-function Identities (Image #5): https://docs.google.com/document/d/1ybRVyNDo25rKRhJVBDLO2jNAQiBi7IuvoNgSYJbJbbQ/edit?usp=sharing
Even/Odd Identities and Pythagorean Identities (Image #6): https://docs.google.com/document/d/1ybRVyNDo25rKRhJVBDLO2jNAQiBi7IuvoNgSYJbJbbQ/edit?usp=sharing
Adding and Subtracting Identities (Image #7): https://docs.google.com/document/d/1ybRVyNDo25rKRhJVBDLO2jNAQiBi7IuvoNgSYJbJbbQ/edit?usp=sharing
Double angle and power reducing formulas (Image #8): https://docs.google.com/document/d/1ybRVyNDo25rKRhJVBDLO2jNAQiBi7IuvoNgSYJbJbbQ/edit?usp=sharing
Product-sum-identities (Image #9): https://docs.google.com/document/d/1ybRVyNDo25rKRhJVBDLO2jNAQiBi7IuvoNgSYJbJbbQ/edit?usp=sharing
Sum-Identities Identities (Image #10): https://docs.google.com/document/d/1ybRVyNDo25rKRhJVBDLO2jNAQiBi7IuvoNgSYJbJbbQ/edit?usp=sharing
Half-Angle Identities (Image #11): https://docs.google.com/document/d/1ybRVyNDo25rKRhJVBDLO2jNAQiBi7IuvoNgSYJbJbbQ/edit?usp=sharing
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
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)
Exploring Equivalent Trigonometric Functions
There are many different formulas that are covered in this topic and we use all of them when we are proving identities
As stated by the name topic it covers how different trigonometric functions are equal.
Chapter 6
Sin, Cos and Tan Graph
Images #2 and #3 on this document: https://docs.google.com/document/d/1ybRVyNDo25rKRhJVBDLO2jNAQiBi7IuvoNgSYJbJbbQ/edit?usp=sharing
Degrees and Radians Wheel
Image #1 on this document: https://docs.google.com/document/d/1ybRVyNDo25rKRhJVBDLO2jNAQiBi7IuvoNgSYJbJbbQ/edit?usp=sharing
Cast rule
• In quadrant 4, only Cos (C) is positive, since x is positive and y is negative.
• In quadrant 3, only Tangent (T) is positive because both x and y are negative.
• In quadrant 2, only Sine (S) is positive, since x is negative and y is positive.
• In quadrant 1, All (A) ratios are positive because both x and y are positive.
Special Triangles With Radians and Degrees and on the Cartesian Plane
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
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°
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
Degrees
The formula to convert degrees into radians is 180°/π
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°.
Radians
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
The formula to convert radians into degrees is to multiply your value by by . π/180°
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.
