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Trigonometric functions

Graphs of sec x, cosec x and cot x

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Further kinematics

Vector methods with projectiles

Variable acceleration in one dimension

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Functions of time: Displacement, velocity, acceleration

Using differentiation to find velocity and acceleration

Maxima and minima problems

Using integration to find velocity and displacement

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Constant acceleration

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Constant acceleration: Deriving formulae

Constant acceleration formulae: SUVAT

Vertical motion under gravity

Modelling in mechanics

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Quantities and units in mechanics

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Normal distribution

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The normal distribution: Finding probabilities

The inverse normal distribution function

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Finding the mean and standard deviation

Normal approximation to the binomial distribution

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Hypothesis testing I

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Venn diagrams

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Vectors II

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Vectors in 3D

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Applications to mechanics: 3D vectors

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Integrating standard functions

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Differentiating sin x and cos x

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Parametric differentiation

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Second derivatives: Concave and convex functions

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Trigonometry and modelling

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Solving trigonometric equations: Double angle formula

Simplifying a cos x + b sin x

Proving trigonometric identities

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Trigonometric functions

Secant, cosecant and cotangent

Graphs of sec x, cosec x and cot x

Solving equations with sec x, cosec x and cot x

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Sequences revision

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Binomial expansion II

Binomial expansion: (1+x)^n

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The modulus function

y = |f(x)| and y = f(|x|)

Combining transformations

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Modulus inequalities

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Proof II

Proof by contradiction

Criticising proofs

Vectors I

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Integration I

Integrating x^n

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Differentiation I

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Differentiating x^n

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Gradients, tangents and normals

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y = e^x

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Trig graphs: sin x, cos x, tan x

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Pascal's triangle

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y = mx + c

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Proof I

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Disproof by counterexample

Proof by deduction

Proof by exhaustion

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Summary

Graphs of $\sec(x)$ , $\cosec(x)$ and $\cot(x)$

In a nutshell

By using the graphs of $y=\sin(x)$ , $y=\cos(x)$ and $y=\tan(x)$ , you can sketch a useful plot of the reciprocal graphs $y=\cosec(x)$ , $y=\sec(x)$ and $y=\cot(x)$ .

The functions

Recall that

$\begin{aligned}\cosec(x)&=\frac1{\sin(x)}\\\sec(x)&=\frac1{\cos(x)}\\\cot(x)&=\frac1{\tan(x)}\end{aligned}$

This helps when using the plots of the graphs of $y=\sin(x)$ , $y=\cos(x)$ and $y=\tan(x)$ .

The graphs

When a graph has an $x$ -intercept, it follows that the reciprocal of that graph has an asymptote, and vice versa. Below, the graphs are given using degrees. The shapes would be the same for radians, but the $x$ -axes would be in radians.

$y=\cosec(x)$

Maths; Trigonometric functions; KS5 Year 13; Graphs of sec x, cosec x and cot x

The domain of this function is all the real numbers, excluding integer multiples of $180$ (in degrees) or multiples of $\pi$ (in radians). At these points, there are asymptotes. The range is $y\leq-1$ or $1\leq y$ . As with $y=\sin(x)$ , it has a period of $360$ (if in degrees) or $2\pi$ (if in radians). Notice that if you drew $y=\sin(x)$ on the same grid, the peaks of $y=\cosec(x)$ would meet the troughs of $y=\sin(x)$ , and vice versa.

$y=\sec(x)$

The domain of this function is all the real numbers, excluding odd integer multiples of $90$ (in degrees) or odd multiples of $\frac{\pi}2$ (in radians). At these points, there are asymptotes. The range is $y\leq-1$ or $1\leq y$ . As with $y=\cos(x)$ , it has a period of $360$ (if in degrees) or $2\pi$ (if in radians). Notice that if you drew $y=\cos(x)$ on the same grid, the peaks of $y=\sec(x)$ would meet the troughs of $y=\cos(x)$ , and vice versa.

$y=\cot(x)$

The domain of this function is all all real numbers, excluding integer multiples of $180$ (in degrees) or integer multiples of $\pi$ (in radians): this is where there are asymptotes. The range is all real numbers.

Learn with Basics

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Unit 1

Trigonometric graphs - Higher

Unit 2

Trig graphs: sin x, cos x, tan x

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