Educational

Sketch the Graph of f(x) = cos πx Question: Sketch the graph of the following trigonometric function : \[ f(x)=\cos \pi x \] Solution: We know that \[ y=\cos x \] is the standard cosine curve. In the function \[ y=\cos \pi x \] the angle is multiplied by \(\pi\). Therefore the graph oscillates faster. […]

Sketch the Graph of u(x) = cos²(x/2) Question: Sketch the graph of the following trigonometric function : \[ u(x)=\cos^2\frac{x}{2} \] Solution: We know that \[ u(x)=\cos^2\frac{x}{2} =\left(\cos\frac{x}{2}\right)^2 \] Since square of cosine is always non-negative, the graph always lies above the x-axis. Using the identity \[ \cos^2\theta=\frac{1+\cos2\theta}{2} \] we get \[ \cos^2\frac{x}{2} =\frac{1+\cos x}{2} \]

Sketch the Graph of ψ(x) = cos 3x Question: Sketch the graph of the following trigonometric function : \[ \psi(x)=\cos3x \] Solution: We know that \[ y=\cos x \] is the standard cosine curve. In the function \[ y=\cos3x \] the angle is multiplied by \(3\). Therefore the graph oscillates faster. Important properties: Amplitude \(=1\)

Sketch the Graph of Φ(x) = 2 cos(x − π/6) Question: Sketch the graph of the following trigonometric function : \[ \Phi(x)=2\cos\left(x-\frac{\pi}{6}\right) \] Solution: We know that \[ y=\cos x \] is the standard cosine curve. The graph of \[ y=2\cos\left(x-\frac{\pi}{6}\right) \] is obtained by: Stretching the cosine curve vertically by factor \(2\) Shifting it

Sketch the Graph of h(x) = cos²2x Question: Sketch the graph of the following trigonometric function : \[ h(x)=\cos^22x \] Solution: We know that \[ h(x)=\cos^22x=(\cos2x)^2 \] Since square of cosine is always non-negative, the graph always lies above the x-axis. Using the identity \[ \cos^2\theta=\frac{1+\cos2\theta}{2} \] we get \[ \cos^22x=\frac{1+\cos4x}{2} \] Hence the period

Sketch the Graph of g(x) = cos(x + π/4) Question: Sketch the graph of the following trigonometric function : \[ g(x)=\cos\left(x+\frac{\pi}{4}\right) \] Solution: We know that \[ y=\cos x \] is the standard cosine curve. The graph of \[ y=\cos\left(x+\frac{\pi}{4}\right) \] is obtained by shifting the graph of \[ y=\cos x \] to the left

Sketch the Graph of f(x) = cos(x − π/4) Question: Sketch the graph of the following trigonometric function : \[ f(x)=\cos\left(x-\frac{\pi}{4}\right) \] Solution: We know that \[ y=\cos x \] is the standard cosine curve. The graph of \[ y=\cos\left(x-\frac{\pi}{4}\right) \] is obtained by shifting the graph of \[ y=\cos x \] to the right

Sketch the Graphs of y = sin(x/2) and y = sin x on the Same Axes Question: Sketch the graphs of the following pairs of functions on the same axes: \[ f(x)=\sin\frac{x}{2} \] \[ g(x)=\sin x \] Solution: We know that \[ y=\sin x \] is the standard sine curve having period \[ 2\pi \]

Sketch the Graphs of y = sin 2x and y = 2 sin x on the Same Axes Question: Sketch the graphs of the following pairs of functions on the same axes: \[ f(x)=\sin 2x \] \[ g(x)=2\sin x \] Solution: Consider the graph of \[ y=\sin 2x \] Its amplitude is \[ 1 \]

Sketch the Graphs of y = sin x and y = sin 2x on the Same Axes Question: Sketch the graphs of the following pairs of functions on the same axes: \[ f(x)=\sin x \] \[ g(x)=\sin 2x \] Solution: We know that \[ y=\sin x \] is the standard sine curve having period \[