Podcast
Questions and Answers
Which of the following is the correct expansion of $\sin(A + B)$?
Which of the following is the correct expansion of $\sin(A + B)$?
- $\cos A \cos B + \sin A \sin B$
- $\sin A \cos B - \cos A \sin B$
- $\sin A \cos B + \cos A \sin B$ (correct)
- $\cos A \cos B - \sin A \sin B$
$\cos(A - B)$ is equivalent to $\cos A \cos B - \sin A \sin B$.
$\cos(A - B)$ is equivalent to $\cos A \cos B - \sin A \sin B$.
False (B)
What is the formula for $\tan(A + B)$?
What is the formula for $\tan(A + B)$?
$\frac{\tan A + \tan B}{1 - \tan A \tan B}$
The double angle formula for sine is: $\sin 2\theta = 2 \sin \theta \cos$ ______
The double angle formula for sine is: $\sin 2\theta = 2 \sin \theta \cos$ ______
Match the following trigonometric identities with their expansions:
Match the following trigonometric identities with their expansions:
Which of these is a correct formula for $\cos 2\theta$?
Which of these is a correct formula for $\cos 2\theta$?
$\tan 2\theta = \frac{2 \tan \theta}{1 + \tan^2 \theta}$ is the correct double angle formula for tangent.
$\tan 2\theta = \frac{2 \tan \theta}{1 + \tan^2 \theta}$ is the correct double angle formula for tangent.
What is the expression for $1 + \cos 2\theta$ using the double angle formulas?
What is the expression for $1 + \cos 2\theta$ using the double angle formulas?
The triple angle formula for $\sin 3\theta$ can be expressed as $3 \sin \theta - 4 \sin^3$ ______
The triple angle formula for $\sin 3\theta$ can be expressed as $3 \sin \theta - 4 \sin^3$ ______
If $f(x) = \tan^{-1}(x)$, what is $f'(x)$?
If $f(x) = \tan^{-1}(x)$, what is $f'(x)$?
Flashcards
sin(A+B) = ?
sin(A+B) = ?
sin(A)cos(B) + cos(A)sin(B)
sin(A-B) = ?
sin(A-B) = ?
sin(A)cos(B) - cos(A)sin(B)
cos(A+B) = ?
cos(A+B) = ?
cos(A)cos(B) - sin(A)sin(B)
cos(A-B) = ?
cos(A-B) = ?
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tan(A+B) = ?
tan(A+B) = ?
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tan(A-B) = ?
tan(A-B) = ?
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sin(C) + sin(D) = ?
sin(C) + sin(D) = ?
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sin(C) - sin(D) = ?
sin(C) - sin(D) = ?
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cos(2θ) = ?
cos(2θ) = ?
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tan(2θ) = ?
tan(2θ) = ?
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Study Notes
Addition Formulae
- sin(A + B) = sinA cosB + cosA sinB
- sin(A - B) = sinA cosB - cosA sinB
- cos(A + B) = cosA cosB + (- sinA sinB)
- cos(A - B) = cosA cosB + sinA sinB
- tan(A + B) = (tanA + tanB) / (1 - tanA tanB)
- tan(A - B) = (tanA - tanB) / (1 + tanA tanB)
Factorization Formulae
- sinC + sinD = 2 sin((C + D)/2) cos((C - D)/2)
- sinC - sinD = 2 cos((C + D)/2) sin((C - D)/2)
- cosC + cosD = 2 cos((C + D)/2) cos((C - D)/2)
- cosC - cosD = -2 sin((C + D)/2) sin((C - D)/2)
Double Angle Formulae
- sin2θ = 2 sinθ cosθ
- cos2θ = cos²θ - sin²θ
- cos2θ = 2 cos²θ - 1
- cos2θ = 1 - 2 sin²θ
- tan2θ = (2 tanθ) / (1 - tan²θ)
- sin2θ = (2 tanθ) / (1 + tan²θ)
- cos2θ = (1 - tan²θ) / (1 + tan²θ)
Important Formulae
- 1 + cos2θ = 2 cos²θ
- 1 - cos2θ = 2 sin²θ
Triple Angle Formulae
- sin3θ = 3 sinθ - 4 sin³θ
- cos3θ = 4 cos³θ - 3 cosθ
- tan3θ = (3 tanθ - tan³θ) / (1 - 3 tan²θ)
- sin3θ / tan3θ = 3 sinθ - sin³θ
- cos3θ = cos³θ - cos³θ + 3θi
Half Angle Formulae
- sin(θ/2) = 2 sin(θ/2) + cos(θ/2)
- cos(θ/2) = cos²(θ/2) - sin²(θ/2)
- cos(θ/2) = 2 cos²(θ/2) - 1
- cos(θ/2) = cos(θ/2) - 1 - 2 sin²(θ/2)
- tan(θ/2) = (2 tan(θ/2)) / (1 - tan²(θ/2))
- sinθ = (2 tan(θ/2)) / (1 + tan²(θ/2))
- cosθ = (1 - tan²(θ/2)) / (1 + tan²(θ/2))
- 1 + sin²θ = (cosθ + sinθ)²
- (1 - cosθ) / (1 + cosθ) = tan²(θ/2)
- (1 + tanθ) / (1 - tanθ) = tan(π/4 + θ)
Inverse Trigonometric Functions
- sin⁻¹(sinx) = x
- cos⁻¹(cosx) = x
- tan⁻¹(tanx) = x
- sin(sin⁻¹x) = x
- cos(cos⁻¹x) = x
- tan(tan⁻¹x) = x
- sin⁻¹(1/x) = cosec⁻¹(x)
- tan⁻¹x = 1 / (1 + x²)
- cot⁻¹x = 1 / (1 + x²)
- sec⁻¹x = 1/(x √(x²-1))
- cosec⁻¹x = -1/(x √(x²-1))
Derivatives
- f(x) = y
- The derivative of xⁿ is nxⁿ⁻¹
- The derivative of 1/x is -1/x²
- The derivative of eˣ is eˣ
- The derivative of eᵃ⁺ᵇ is eᵃ⁺ᵇ(a)
- The derivative of log x is 1/x
- The derivative of logₐx is 1/(x log a)
- The derivative of aˣ is aˣ log a
- The derivative of aˣ/(b+c) is (aˣ log a)/(b+c)
- The derivative of a constant is 0
- √(x) the derivative is equal to 1/(2√(x))
- derivative of sinx = cosx
- derivative of sin(ax+b) = cos(ax+b)a
- derivative of cosx = -sinx
- derivative of cos(ax+b) = -sin(ax+b)a
- derivative of tanx = sec²x
- derivative of tan(ax+b) = sec²(ax+b)a
- derivative of cotx = -cosec²x
- derivative of secx = secxtanx
- derivative of sec(ax+b) = (sec(ax+b)tan(ax+b))a
- derivative of cosecx = -cosecxcotx
- derivative of cosec(ax+b) = (-cosec(ax+b)cot(ax+b))a
- derivative of sin⁻¹(x) = 1/√(1-x²)
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