Force

Trigonometry

Unit: 7

Book Icon Class 10: Optional Math

Trigonometric Relations of Multiple Angles, Trigonometric Relations of Sub-Multiple Angles, Transformation of Trigonometric Ratios, Conditional Trigonometric Identities, Trigonometric Equation, Height and Distance

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

\(\begin{aligned} & 1. sin2A = 2sinAcosA \\ & 2. cos2A = cos^{2}A – sin^{2}A, 2cos^{2}A – 1, 1 – 2sin^{2}A \\ & 3. tan2A = \frac{2tanA}{1 – tan^{2}A} \\ & 4. sin3A = 3sinA – 4sin^{3}A \\ & 5. cos3A = 4cos^{3}A – 3cosA \\ & 6. tan3A = \frac{3tanA – tan^{3}A}{1-3tan^{2}A} \end{aligned}\)

 

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.

\(\begin{aligned} & 1. sinC + sinD = 2sin(\frac{C+D}{2})cos(\frac{C-D}{2}) \\ & 2. sinC - sinD = 2cos(\frac{C+D}{2})sin(\frac{C-D}{2}) \\ & 3. cosC + cosD = 2cos(\frac{C+D}{2})cos(\frac{C-D}{2}) \\ & 4. cosC - cosD = 2sin(\frac{C+D}{2})sin(\frac{C-D}{2}) \\ \end{aligned}\)

 

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