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\( \begin{array}{ll}\text { (1) } \frac{\sin 2 x}{2 \cos ^{2} x} & \text { (2) } \frac{\cos 2 \theta}{\sin \theta+\cos \theta} \\ \begin{array}{ll}\text { (4) } \frac{(\cos \alpha-\sin \alpha)^{2}}{\sin 2 \alpha-1} & \text { (5) } \frac{\cos 2 x}{\cos ^{4} x-\sin ^{4} x}\end{array} \\ \text { (d)* Calculate the following without the use of a calculator: } \frac{1-\cos 2 \mathrm{~A}}{1-\cos ^{2} \mathrm{~A}} \frac{\sin 39^{\circ}}{\sin 13^{\circ}}-\frac{\cos 39^{\circ}}{\cos 13^{\circ}}\end{array} \)

Ask by Elliott Pearson. in South Africa
Mar 10,2025

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(1) \( \frac{\sin 2x}{2 \cos^2 x} = \tan x \) (2) \( \frac{\cos 2\theta}{\sin \theta + \cos \theta} = \cos \theta - \sin \theta \) (4) \( \frac{(\cos \alpha - \sin \alpha)^2}{\sin 2\alpha - 1} = \frac{\cos^2 \alpha - 2 \sin \alpha \cos \alpha + \sin^2 \alpha}{2 \sin \alpha \cos \alpha - 1} \)

Solución

(1) We start with \[ \frac{\sin2x}{2\cos^2x}. \] Recall that \[ \sin2x=2\sin x\cos x. \] Thus, \[ \frac{\sin2x}{2\cos^2x}=\frac{2\sin x\cos x}{2\cos^2x}=\frac{\sin x}{\cos x}=\tan x. \] (2) Consider \[ \frac{\cos2\theta}{\sin\theta+\cos\theta}. \] Write the numerator in factorized form. Notice that \[ \cos2\theta=\cos^2\theta-\sin^2\theta=(\cos\theta+\sin\theta)(\cos\theta-\sin\theta). \] Then \[ \frac{\cos2\theta}{\sin\theta+\cos\theta} =\frac{(\cos\theta+\sin\theta)(\cos\theta-\sin\theta)}{\sin\theta+\cos\theta} =\cos\theta-\sin\theta, \] provided \(\sin\theta+\cos\theta\neq 0\). (4) Next, \[ \frac{(\cos\alpha-\sin\alpha)^2}{\sin2\alpha-1}. \] First, expand the numerator: \[ (\cos\alpha-\sin\alpha)^2=\cos^2\alpha-2\sin\

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The Deep Dive

Did you know that the sine and cosine functions were used by ancient civilizations long before they were fully understood as we know them today? The Babylonians had a method of calculating angles and distances using these functions over 4,000 years ago. They didn’t have calculators or modern mathematics, yet they were able to create pyramids and other monumental structures using these early concepts of trigonometry. In terms of real-world application, trigonometry is the backbone of various fields including engineering, architecture, and even video game design! For example, in architecture, architects use sine and cosine functions to calculate structural angles and create stable designs. Even in animation, trigonometric functions help in creating realistic movements and rotations. Who knew math could be so cool and versatile?

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