Trigonometry flashcards

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What is the definition of sec(x)?

sec(x) is the secant of x, defined as 1/cos(x). It is undefined when cos(x) = 0.

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What is the definition of csc(x) (or cosec(x))?

csc(x) is the cosecant of x, defined as 1/sin(x). It is undefined when sin(x) = 0.

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What is the definition of cot(x)?

cot(x) is the cotangent of x, defined as 1/tan(x), which is equivalent to cos(x)/sin(x). It is undefined when sin(x) = 0.

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State the Pythagorean identity involving sec(x).

1 + tan²(x) ≡ sec²(x). This is derived by dividing sin²(x) + cos²(x) ≡ 1 by cos²(x).

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State the Pythagorean identity involving cot(x).

1 + cot²(x) ≡ csc²(x). This is derived by dividing sin²(x) + cos²(x) ≡ 1 by sin²(x).

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What are the three forms of the double angle formula for cos(2A)?

1. cos²(A) - sin²(A) 2. 2cos²(A) - 1 3. 1 - 2sin²(A)

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When expressing a sin(θ) + b cos(θ) as R sin(θ + α), what are the formulae for R and α?

R = √(a² + b²) and tan α = b/a

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What is the principal value range for y = arcsin(x)?

-π/2 ≤ y ≤ π/2 (or -90° ≤ y ≤ 90°). The output is in quadrants 1 and 4.

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What is the principal value range for y = arccos(x)?

0 ≤ y ≤ π (or 0° ≤ y ≤ 180°). The output is in quadrants 1 and 2.

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What is the principal value range for y = arctan(x)?

-π/2 < y < π/2 (or -90° < y < 90°). The output is in quadrants 1 and 4.

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Common mistake: When solving tan²(x) = 3, what is a frequently missed step?

Forgetting the ± sign when taking the square root. tan(x) = ±√3, leading to solutions in all four quadrants.

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Secant

The reciprocal of cosine: $\sec x = \dfrac{1}{\cos x}$. Undefined wherever $\cos x = 0$. Related to tangent by the Pythagorean identity $1 + \tan^2 x \equiv \sec^2 x$.

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Cosecant

The reciprocal of sine: $\operatorname{cosec} x = \dfrac{1}{\sin x}$ (also written $\csc x$). Undefined wherever $\sin x = 0$. Related to cotangent by $1 + \cot^2 x \equiv \operatorname{cosec}^2 x$.

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Cotangent

The reciprocal of tangent: $\cot x = \dfrac{1}{\tan x} = \dfrac{\cos x}{\sin x}$. Undefined wherever $\sin x = 0$.

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Compound angle formulae

The expansions of trigonometric functions of a sum or difference of angles, such as $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$ and $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$. Setting $B = A$ in them gives the double angle formulae.

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Double angle formulae

The expansions of $\sin 2A$, $\cos 2A$ and $\tan 2A$ obtained by putting $B = A$ in the compound angle formulae: $\sin 2A = 2\sin A \cos A$, $\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$.

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Harmonic form

Writing $a\sin x + b\cos x$ as a single trigonometric function, $R\sin(x + \alpha)$ or $R\cos(x - \alpha)$, with $R = \sqrt{a^2 + b^2}$. It makes equations of the form $a\sin x + b\cos x = c$ easy to solve and gives the maximum ($R$) and minimum ($-R$) of the expression directly.