A-Level AQA Maths: Pure 5.2 Trigonometric Functions: Using small angle approximations

Using small angle approximations, show that for small, non-zero, values of $x$ $$\frac{x \tan 5x}{\cos 4x - 1} \approx A$$ where $A$ is a constant to be determined. - AQA - A-Level Maths: Pure - Question 4 - 2020 - Paper 2

Question 4

$Using-small-angle-approximations,-show-that-for-small,-non-zero,-values-of-$x$--$$\frac{x-\tan-5x}{\cos-4x---1}-\approx-A$$--where-$A$-is-a-constant-to-be-determined.-AQA-A-Level Maths: Pure-Question 4-2020-Paper 2.png$

Using small angle approximations, show that for small, non-zero, values of $x$ $$\frac{x \tan 5x}{\cos 4x - 1} \approx A$$ where $A$ is a constant to be determined... show full transcript

Worked Solution & Example Answer:Using small angle approximations, show that for small, non-zero, values of $x$ $$\frac{x \tan 5x}{\cos 4x - 1} \approx A$$ where $A$ is a constant to be determined. - AQA - A-Level Maths: Pure - Question 4 - 2020 - Paper 2

Step 1

Using small angle approximation for $\tan 5x$

96%

114 rated

Answer

For small angles, we can use the approximation: $\tan \theta \approx \theta.$ Thus, for $\tan 5x$ , we have $\tan 5x \approx 5x.$

Step 2

Using small angle approximation for $\cos 4x$

99%

104 rated

Answer

Using the small angle approximation for cosine, we know that: $\cos \theta \approx 1 - \frac{\theta^2}{2}.$ Therefore, for $\cos 4x$ , we can write: $\cos 4x \approx 1 - \frac{(4x)^2}{2} = 1 - 8x^2.$

Step 3

Substituting into the expression

96%

101 rated

Answer

We substitute these approximations into our expression: $\frac{x \tan 5x}{\cos 4x - 1} \approx \frac{x (5x)}{(1 - 8x^2) - 1} = \frac{5x^2}{-8x^2} = -\frac{5}{8}.$ Thus, we can conclude that $A = -\frac{5}{8}$ .

Step 4

Final deduction of constant $A$

98%

120 rated

Answer

Finally, we have shown that for small values of $x$ , $\frac{x \tan 5x}{\cos 4x - 1} \approx A$ where $A = -\frac{5}{8}.$

Using small angle approximations, show that for small, non-zero, values of $x$ $$\frac{x \tan 5x}{\cos 4x - 1} \approx A$$ where $A$ is a constant to be determined. - AQA - A-Level Maths: Pure - Question 4 - 2020 - Paper 2

Worked Solution & Example Answer:Using small angle approximations, show that for small, non-zero, values of $x$ $$\frac{x \tan 5x}{\cos 4x - 1} \approx A$$ where $A$ is a constant to be determined. - AQA - A-Level Maths: Pure - Question 4 - 2020 - Paper 2

Using small angle approximation for $\tan 5x$

Using small angle approximation for $\cos 4x$

Substituting into the expression

Final deduction of constant $A$

Join the A-Level students using SimpleStudy...

Other A-Level Maths: Pure topics to explore