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Trigonometry: Key Concepts - Coggle Diagram
Trigonometry: Key Concepts
Trigonometry 1
Radian Measure
angles can be measured in radians
a radian is the measure of the angle at the centre of a circle subtended by an arc equal in length to the radius
LOG TABLES
π radians = 180 degrees
2π radians = 360 degrees
arc length L = rΘ
Θ = theta = unknown/unspecified angle
area of a sector = 1/2r^2Θ
Trig. Ratios
SOH CAH TOA
Sin A = opposite side/hypotenuse
Cos A = adjacent side/hypotenuse
Tan A = opposite side/adjacent side
pg 13 of Log Tables
Trig. Functions
The Unit Circle
radius 1 unit
centre at (0,0)
Four Quadrants
The reference angle, of any angle A, is always the smallest (acute) angle to the horizontal x-axis
USE CAST CIRCLE
Sine Rule
Log Tables
a/sinA = b/sinB = c/SinC
(WORKS INVERSELY)
Used to find angles and sides of a non-right angle triangle
PROOF
To prove the Sine Rule with an acute angled triangle, establish that h, the height from base to top vertex, over b (any side) = Sin A, which means h = bSinA
Similarly, h/a (other side) = sin B, meaning h = asinB
∴ a sin B = b Sin A
Area of a Triangle
area = half the product of any two sides multiplied by the sine of the angle between them
Cosine Rule
Log Tables
a^2 = b^2 + c^2 - 2bcCosA
PROOF
Draw ∆ with vertex A at the origin, B(c,0) and point C (making an obtuse angle with A)
Draw a circle of radius length b with centre at A
Since the coords. of any point on a unit circle can be written as (cosΘ,sinΘ), the coords. of C can be written as (bCosA,bSinA)
|BC| = a and |BC| = distance from (c,0) to (bCosA,bSinA
)
a^2 = (c - bcosA)^2 + (0-bSinA)^2
a^2 = c^2 - 2bcCosA + b^2cos^2A + b^2Sin^2A
= c^2 - 2bcCosA + b^2 (cos^2A + Sin^2A)
= c^2 - 2bcCosA + b^2
---> a^2 = b^2 + c^2 - 2bcCosA
3-D Trig.
If a triangle isn't right-angled, use cosine or sine rule to find uncommon angle/s or side/s
Graphs of Trig. Functions
The period of a function, f, is a positive number a such that f(x + a) = f(x)
The period of sin(x) is 2π as sin(x + 2π) = sin(x)
Understand amplitude, range, period, asymptote, domain, mid-line and max/min
General Solutions of Trig. Equations
check that Θ = 300 degrees and Θ = 420 degrees also work, as when solving, calculator may only give one principal value
To find the general solution of sin x = k or cos x = k, you find the two solutions in the interval 0 degrees ≤ Θ ≤ 360 degrees and then add n360 degrees to each of the solutions
Trigonometry 2