Trigonometric identities for OCR A-level Maths
This page covers the following topics:
1. Trigonometric identities
2. Double angle formulae
3. Adding trigonometric functions
4. Trigonometric equations
5. Small angle approximations
Two commonly used trigonometric identities are tanx = sinx/cosx and sin²x + cos²x = 1. They can be used to manipulate and simplify trigonometric expressions. The ratio of sine and cosine identity is useful when the ratio of two functions is obtained or when a tangent function is worth expanding. The sum of sine squared and cosine squared can be applied to simplify an expression containing both of the squares.
Double angle formulae can be used in simplifying and solving trigonometric equations. The three main formulae are sin(A + B) = sinAcosB + cosAsinB, cos(A + B) = cosAcosB − sinAsinB and tan(A + B) = (tanA + tanB) ÷ (1 − tanAtanB). The double angle formula for tangent is derived by combining the ones for sine and cosine and dividing every term by cosAcosB. Formulae for sin(A − B), cos(A − B) and tan(A − B) can be derived from these, using the fact that cos(−x) = cosx and sin(−x) = −sinx. The following formulae can be derived from the double angle formulae: sin(2x) = 2sinxcosx, cos2x = cos²x − sin²x = 1 - 2sin²x = 2cos²x − 1 and tan2x = 2tanx ÷ (1 − tan²x).
Addition formulae can be used to express combinations of sine and cosine expressions into a single trigonometric function. Expressions in the form asinx ± bcosx can be expressed as Rsin(x ± θ). Expressions in the form acosx ± bsinx can expressed as Rcos(x ∓ θ). R and θ can be found using the Pythagoras theorem and tangent identity using the double angle formulae. R is the maximum value the expression can take and −R is the minimum value it can take.
For example, asinx + bcosx can be expressed in the form Rsin(x ± θ). By the appropriate double angle formula, Rsin(x + θ) = Rsinxcosθ + Rcosxsinθ. Comparing this to the initial expression, the following equations are derived: a = Rcosθ and b = Rsinθ. Thus, b ÷ a = tanθ, and θ can be calculated using the inverse tangent function. Also, squaring and adding these expressions gives a² + b² = R²cos²θ + R²sin²θ = R²(cos²θ + sin²θ) = R² and R = √(a² + b²).
The trigonometric additional formulae and double angle formulae can be used in solving trigonometric equations. When multiple trigonometric functions are involved in an equation, they can sometimes be combined together to a single function that can simplify obtaining an unknown variable.
Small angle approximations are used to simplify problems. Small angle approximations are estimates that can be made when the angle θ is small. sinθ can be approximated by θ, cosθ can be approximated by 1 − θ²/2 and tanθ can be approximated by θ. Plotting the graphs of these approximations on the same axes as the graphs for the trigonometric functions shows that for small values of θ, the two graphs overlap, which is why these approximations hold.
Give the following expression only in terms of cosine: √3cosx/2 + sinx/2.
The formula cos(A − B) = cosAcosB + sinAsinB is used.
√3cosx/2 + sinx/2 =
= cos30°cosx + sin30°sinx =
= cos(30° − x)
cos(30° − x)
Use the addition formula for sine to derive a formula for sin(A − B).
The formula sin(A + B) = sinAcosB + cosAsinB is used.
Let B = −B.
sin(A − B) = sinAcos(−B) + cosAsin(−B)
cos(−x) = cosx and sin(−x) = −sinx are now used.
sin(A − B) = sinAcosB − cosAsinB
sin(A − B) = sinAcosB − cosAsinB
Use trigonometric identities to give the following expression in terms of cosx: (sinx) ÷ (tanx) + sin²x.
The identities tanx = sinx/cosx and sin²x + cos²x = 1 are used.
(sinx) ÷ (tanx) + sin²x =
= cosxsinx ÷ sinx + sin²x =
= cosx + sin²x =
= cosx + 1 − cos²x
cosx + 1 − cos²x
Simplify the following expression: sin(x + y) − sin(x − y).
The addition formulae for sine can be used.
sin(x + y) − sin(x − y) =
= sinxcosy + cosxsiny − sinxcosy + cosxsiny =
Rewrite the following expression only in terms of sine: 8sinx − 12cosx.
The expression can be rewritten in the form Rsin(x − θ).
R = √(8² + 12²) = 4√13
θ = arctan(12 ÷ 8) = 56.3° (to 3 s. f.)
4√13sin(x − 56.3°)
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