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1. Tick () the correct option for the given questions:
A. Which of the following statements is incorrect regarding the composite functions?
a. If functions f: A → B and g: B → C exist, and the relation A → C is also defined as a function, it is a composite function.
b. If functions g: A → B and f: B → C are one to one and onto, then, the function A → C is a composite function.
c. If functions g: A → B and f: B → C are defined, the function A → C is denoted by g∘f.
d. If functions f: A → B and g: B → C are defined, the relation C → A cannot be a composite function of f and g.

Solution:
C

 

1. (b) Write \( \cos2A \) in terms of \( \cos A \) and \( \sin A \)

Solution:

\[ \cos2A = 2\cos^2 A-1 \]

\[ \cos2A = 1- 2\sin^2 A \]

 

2. (a) Write \( \sin 2A \) in terms of \( \tan A \)

Solution:

\[ \sin 2A = \frac{2 \tan A}{ 1+ \tan^2 A } \]

 

2(b) Write \( \tan^2 A \) in terms of \( \cos2A \)

Solution:

\[ \tan^2 A = \frac{1-\cos 2 A }{1+ \cos 2 A } \]

2(c) Write \( \sin3A \) in terms of \( \sin A \).

Solution:

\[ \sin 3A = 3\sin A - 4\sin^3 A \]

2(d) Write \( \tan3θ \) in terms of \( \tan \theta \)

Solution:

\[ \tan 3\theta = \frac{ 3 \tan \theta -\tan^3 \theta}{1-3\tan^2 \theta} \]

3.(a) If \( \sin A = \frac{3}{5} \), find the value of \( \cos 2A \).

Solution:

Given,

\[ \sin A =\frac{3}{5} \]

Now,

\begin{align} \cos 2A &=1-2\sin^2 A\\ \ &=1-2\times\left(\frac{3}{5} \right)^2\\ \ &= 1-\frac{18}{25}\\ \ &=\frac{25-18}{25}\\ \ &=\frac{7}{25}\\ \therefore \cos 2A &= \frac{7}{25} \end{align}

 

3. (b) If \( \sin 2A = \frac{ 24}{25} \) and \( \sin A = \frac{4}{5} \) , then find the value of \( \cos A \).

Solution:

Given,

\[ \sin 2A = \frac{ 24}{25} \]

and

\[ \sin A = \frac{4}{5} \]

We know that,

\begin{align} \sin 2A &=2\sin A \cos A \\ or,\frac{24}{25}&=2\times \frac{4}{5}\times \cos A\\

All The Best