AM
Animated Mathematics
This is presumed knowledge; typically students know how to calculate an angle given two sides and the length of a side given an angle and a side. We note 0≤cosθ ≤1 and 0≤sinθ≤1 since the adjacent and opposite are always shorter than the hypotenuse. We can introduce some the abstract concept of infinity by reducing the length of the adjacent; in this limit tanθ approaches positive infinity and arctanθ approaches 90 degrees. pdf of notes
a blank space
Introduce radians without explaining why it’s useful, state the simple conversion 180 degrees equals π radians. Calculation of some trigonometric ratios without a calculator, by considering an equilateral triangle and a right angled isosceles triangle. The unit circle is used to defines cosθ and sinθ for θ>90 degrees. There is a point P(x,y) on the unit circle, and cosθ=x and sinθ=y. The trig ratio for 30, 45, and 60 degrees can be found be written down for the first quadrant. The coordinates of P in the other quadrants can be determined for some special angles by considering the coordinates written down for quadrant I. Fill in a unit circle and test your knownledge by filling in this unit circle online.
To solve cos(theta)=k, we consider the intersection points of the unit circle with the vertical line x=k and to solve sin(theta)=k we consider the intersection points of the unit circle with horizontal line y=k. There can be at most two solutions. pdf of notes
Examples of solving sinθ/cosθ/tanθ=k where k is a real number, using the unit circle. Deriving trigonometrical identities by consider quadrant I and then comparing to quadrants II, III, and IV in turn. The arc length formula with the angle in radians.
The sector area formula with the angle in radians. The formula for the area of a triangle in terms of one angle and the length of the sides that define that angle. The formula for the periodicity T of sin(kx) and cos(kx) can be determined by considering a point on the unit circle moving with a constant radial velocity.
Changes in the amplitude, period, of sine and cosine waves. The vertical shift of such functions. Intersection points of sine and cosine functions. Solving trigonometric functions and the graphical interpretation of the solution.