Trigonometric Relations of Multiple Angles
What are multiple angles?
If A is an angle, multiples of A i.e. 2A, 3A, 4A, … etc are called multiple angles of A.
Formulas Related to Multiple Angles
\(\begin{aligned}
& 1. sin2A = 2sinAcosA, \frac{2tanA}{1 + tan^{2}A} \\
& 2. cos2A = cos^{2}A – sin^{2}A, 2cos^{2}A – 1, 1 – 2sin^{2}A \\
& 3. tan2A = \frac{2tanA}{1 – tan^{2}A} \\
& 4. sin3A = 3sinA – 4sin^{3}A \\
& 5. cos3A = 4cos^{3}A – 3cosA \\
& 6. tan3A = \frac{3tanA – tan^{3}A}{1-3tan^{2}A}
\end{aligned}\)
Trigonometric Relations of Sub-Multiple Angles
What are sub-multiple angles?
If \(A\) is an angle, then \(\frac{A}{2}\), \(\frac{A}{3}\), \(\frac{A}{4}\), … \(\frac{A}{n}\) , \(n \in N\) are called multiple angles of \(A\).
Formulas Related to Sub-multiple Angles
\(\begin{aligned}
& 1. sinA = 2sin\frac{A}{2}cos\frac{A}{2} \\
& 2. cosA = cos^{2}\frac{A}{2} – sin^{2}\frac{A}{2}, 2cos^{2}\frac{A}{2} – 1, 1 – 2sin^{2}\frac{A}{2} \\
& 3. tanA = \frac{2tan\frac{A}{2}}{1 – tan^{2}\frac{A}{2}} \\
& 4. sinA = 3sin\frac{A}{3} – 4sin^{3}\frac{A}{3} \\
& 5. cosA = 4cos^{3}\frac{A}{3} – 3cos\frac{A}{3} \\
& 6. tanA = \frac{3tan\frac{A}{3} – tan^{3}\frac{A}{3}}{1-3tan^{2}\frac{A}{3}}
\end{aligned}\)
Transformation of Trigonometric Ratios
The transformation from multiplication to addition or subtraction.
\(\begin{aligned}
& 1. 2sinA.cosB = sin(A+B) + sin(A-B) \\
& 2. 2cosA.sinB = sin(A+B) - sin(A-B) \\
& 3. 2cosA.cosB = cos(A+B) + cos(A-B) \\
& 4. 2sinA.sinB = cos(A-B) - sin(A+B) \\
\end{aligned}\)
The transformation from multiplication to addition or subtraction.
\(\begin{aligned}
& 1. sinC + sinD = 2sin(\frac{C+D}{2})cos(\frac{C-D}{2}) \\
& 2. sinC - sinD = 2cos(\frac{C+D}{2})sin(\frac{C-D}{2}) \\
& 3. cosC + cosD = 2cos(\frac{C+D}{2})cos(\frac{C-D}{2}) \\
& 4. cosC - cosD = 2sin(\frac{C+D}{2})sin(\frac{C-D}{2}) \\
\end{aligned}\)
Trigonometric Equation
A trigonometric equation is an equation that involves trigonometric ratios like sin, cos, tan, etc. and a variable (usually an angle).
Example:
\(\sin x = 0.5\)
This equation means we need to find the values of \(x\) (angle) where \(sin x = 0.5\).
One solution is \(x = 30°\) (or \(π/6\) in radians), but there are more solutions since sine is periodic.
\(\begin{aligned}
& Solve. (0 \leq \theta \leq 360^\circ) \\
& \text{Q. } cos\theta + cos3\theta = 2cos2\theta \\
& \text{Solution: } \\
& \text{or, } cos\theta + 4cos^{3}\theta – 3cos\theta = 2(2cos^{2}\theta - 1) \\
& \text{or, } 4 cos^{3}\theta – 2 cos\theta = 4 cos^{2}\theta - 2 \\
& \text{or, } 4 cos^{3}\theta – 2 cos\theta - 4cos^{2}\theta + 2 = 0 \\
& \text{or, } 4 cos^{3}\theta - 4cos^{2}\theta – 2 cos\theta + 2 = 0 \\
& \text{or, } 4 cos^{2}\theta (cos\theta - 1) - 2 (cos\theta – 1) = 0 \\
& \text{or, } (cos\theta - 1) (4 cos^{2}\theta - 2)= 0 \\
& \text{Either } cos\theta – 1 = 0 \\
& \text{or, } cos\theta = 1 \\
& \text{or, } cos\theta = cos0^\circ, cos360^\circ \\
& \therefore \theta = 0^\circ, 360^\circ \\ \\
& \text{OR, } 4 cos^{2}\theta – 2 = 0 \\
& \text{or, } 4 cos^{2}\theta = 2 \\
& \text{or, } cos^{2}\theta = \frac{1}{2} \\
& \text{or, } cos\theta = \frac{1}{\sqrt{2}} \\
& \text{or, } cos\theta = cos45^\circ, cos(360^\circ – 45^\circ) \\
& \text{or, } cos\theta = cos45^\circ, cos(315^\circ) \\
& \therefore \theta = 45^\circ , 315^\circ \\
& \text{Hence, } \theta = 0^\circ , 45^\circ , 315^\circ , 360^\circ \text{ (Answer)}
\end{aligned}\)
