– Sine, Cosine and Tangent. Sine ruleand cosine rule
Definition : A radian is the angle subtended at the center of a circle by an arc equal to the radius of the circle.
Arc length= rθ
Area of sector= \(\frac{1}{2}θr^2\)
sin(-x)=-sinx
cos(-x)=cosx
tan(-x)=-tanx
tanx = \(\frac{sinx}{cosx}\)
The sine and cosines rule for a triangle is
\(\frac{a}{sinA}\) = \(\frac{b}{sinB}\) = \(\frac{c}{sinC}\)
\(a^2 = b^2 + c^2 - 2bc cosA\)
Area of a triangle = \(\frac{1}{2}\) ab sinC = \(\frac{1}{2}\) bc sinA = \(\frac{1}{2}\) ac sinB
– Small angle approximations
\( sinx = x - \frac{x^3}{3!}\) + \(o(x^3)\)
\( cosx = 1 - \frac{x^2}{2!}\) + \(o(x^2)\)
\( tanx = x + \frac{x^3}{3}\) + \(o(x^3)\)
-Graphs of sine, cosine and tangent
See the graphs
Graph of y = sin3x is like the graph of y = sinx but repeats itself between x = 0 and x =2π
Graph of y = sin(x + 300) is like the graph of but translated by y = sinx but translated by (-300), 0)
– Secant, Cosecant and Cotangent
\(sec x\) = \(\frac{1}{cos x}\) \(cosec x\) = \(\frac{1}{sin x}\) \(cot x\) = \(\frac{1}{tan x}\)
– Trigonometric Identities
\(tan x\) = \(\frac{sin x}{cos x}\)
\(sin^2 x + cos^2x\) = \(1\)
\(tan^2 x + 1\) = \(sec x^2\)
\(1 + cot^2x\) = \(cosec x^2\)
– Double angle formulae
sin(x±y) = sin x cos x ± sin y cos x
sin 2x = 2sin x cos x
cos (x ± y) = cos x cos y ∓ sin x sin y
\(cos 2x\) = \(2cos^2x -1 \) = \(1 - 2sin^2x\)
\(tan 2x\) = \(\frac{2tan x}{1 - tan^2x}\)
– Trigonometric Equations
\(sin(x - \frac{\pi}{4})\) = 0.5
\(sec^2x\) = 3 – tan x
– Proofs of Identities
Prove \(\frac{cos 2x + 1 }{1 - cos 2x}\) =\(cot^2x\)
Rcos(x+α)=Rcos xcosα-Rsinx sin α
– Applications to vectors, kinematics and forces
Dot product is \(\vec{a}\) . \(\vec{b}\) = |a|.|b|.cos ( \(\vec{a}\) \(\vec{b}\))
Work is a dot product
Cross- product is \(\vec{a}\) . \(\vec{b}\) = \(\vec{c}\)
| \(\vec{a}\) . \(\vec{b}\) | = | \(\vec{a}\)| × | \(\vec{b}\)| . sin ( \(\vec{a}\) . \(\vec{b}\) )
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