Educational

Find the Value of tan 20° + tan 40° + √3 tan 20° tan 40° Find the Value of tan 20° + tan 40° + √3 tan 20° tan 40° Question: \[ \tan20^\circ+\tan40^\circ+\sqrt{3}\tan20^\circ\tan40^\circ \] is equal to (a) \(\frac{\sqrt{3}}{4}\) (b) \(\frac{\sqrt{3}}{2}\) (c) \(\sqrt{3}\) (d) \(1\) Solution We use the identity: \[ \tan(A+B) = \frac{\tan A+\tan […]

If A + B + C = π, Find sec A (cos B cos C − sin B sin C) If A + B + C = π, Find sec A (cos B cos C − sin B sin C) Question: If \[ A+B+C=\pi \] then \[ \sec A(\cos B\cos C-\sin B\sin C) \] is

Find the Value of sin²(5π/12) − sin²(π/12) Find the Value of sin²(5π/12) − sin²(π/12) Question: The value of \[ \sin^2\left(\frac{5\pi}{12}\right) – \sin^2\left(\frac{\pi}{12}\right) \] is (a) \(\frac{1}{2}\) (b) \(\frac{\sqrt{3}}{2}\) (c) \(1\) (d) \(0\) Solution We use the identity: \[ \sin^2 A-\sin^2 B = (\sin A-\sin B)(\sin A+\sin B) \] Also, \[ \sin^2 A-\sin^2 B = \sin(A+B)\sin(A-B)

Prove that (2√3 + 3) sin x + 2√3 cos x Lies Between −(2√3 + √15) and (2√3 + √15) Prove that (2√3 + 3) sin x + 2√3 cos x Lies Between −(2√3 + √15) and (2√3 + √15) Question: Prove that \[ (2\sqrt{3}+3)\sin x+2\sqrt{3}\cos x \] lies between \[ -(2\sqrt{3}+\sqrt{15}) \] and \[

Show that sin 100° − sin 10° is Positive Show that sin 100° − sin 10° is Positive Question: Show that \[ \sin100^\circ-\sin10^\circ \] is positive. Solution We use the identity: \[ \sin C-\sin D = 2\cos\left(\frac{C+D}{2}\right) \sin\left(\frac{C-D}{2}\right) \] Taking \[ C=100^\circ, \qquad D=10^\circ \] Then, \[ \sin100^\circ-\sin10^\circ = 2\cos\left(\frac{100^\circ+10^\circ}{2}\right) \sin\left(\frac{100^\circ-10^\circ}{2}\right) \] \[ = 2\cos55^\circ\sin45^\circ

Reduce 24 cos x + 7 sin x to the Sine and Cosine of a Single Expression Reduce 24 cos x + 7 sin x to the Sine and Cosine of a Single Expression Question: Reduce the following expression to the sine and cosine of a single expression: \[ 24\cos x+7\sin x \] Solution We

Reduce cos x − sin x to the Sine and Cosine of a Single Expression Reduce cos x − sin x to the Sine and Cosine of a Single Expression Question: Reduce the following expression to the sine and cosine of a single expression: \[ \cos x-\sin x \] Solution We use the standard form:

Reduce √3 sin x − cos x to the Sine and Cosine of a Single Expression Reduce √3 sin x − cos x to the Sine and Cosine of a Single Expression Question: Reduce the following expression to the sine and cosine of a single expression: \[ \sqrt{3}\sin x-\cos x \] Solution We use the

Find the Maximum and Minimum Value of sin x − cos x + 1 Find the Maximum and Minimum Value of sin x − cos x + 1 Question: Find the maximum and minimum values of the following trigonometrical expression: \[ \sin x-\cos x+1 \] Solution We know that the expression \[ a\sin x+b\cos x

Find the Maximum and Minimum Value of 5 cos x + 3 sin(π/6 − x) + 4 Find the Maximum and Minimum Value of 5 cos x + 3 sin(π/6 − x) + 4 Question: Find the maximum and minimum values of the following trigonometrical expression: \[ 5\cos x + 3\sin\left(\frac{\pi}{6}-x\right)+4 \] Solution Using the