Question

If \(\sec \theta=\frac{\mathrm{a}+\mathrm{b} \cos \phi}{\mathrm{a} \cos \phi+\mathrm{b}}\) then \(\tan \left(\frac{\theta}{2}\right)\) is (a) \(\left(\frac{\mathrm{a}-\mathrm{b}}{\mathrm{a}+\mathrm{b}}\right) \tan \left(\frac{\phi}{2}\right)\) (b) \(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{\phi}{2}\right)\) (c) \(\frac{a b}{a-b} \tan \phi\) (d) \(\frac{\mathrm{a}+\mathrm{b}}{\mathrm{ab}} \cot 2 \phi\)

Short answer

Question: Given that \(\sec(\theta) = \frac{a+b\cos(\phi)}{a\cos(\phi) + b}\), find the expression for \(\tan(\frac{\theta}{2})\). Answer: (b) \(\sqrt{\frac{a-b}{a+b}} \tan\left(\frac{\phi}{2}\right) \).

Step-by-step solution

Manipulate the given relationship

We are given \(\sec(\theta) = \frac{a+b\cos(\phi)}{a\cos(\phi) + b}\). Knowing that \(\sec(\theta)=\frac{1}{\cos(\theta)}\), we can rewrite the equation as: $$\cos(\theta) = \frac{a\cos(\phi) + b}{a+b\cos(\phi)}$$. Now we use the identity \(\cos(2x) = 1 - 2\sin^2(x)\) and \(\cos(\theta)=\cos(2\cdot\frac{\theta}{2})\). So, our equation becomes: $$1 - 2\sin^2(\frac{\theta}{2}) = \frac{a\cos(\phi) + b}{a+b\cos(\phi)}$$.

Find the expression for \(\tan(\frac{\theta}{2})\)

Now we need to derive an expression for \(\tan(\frac{\theta}{2})\). We first manipulate the equation to have the \(\sin^2\) term: $$\sin^2(\frac{\theta}{2}) = \frac{1}{2} \left(1 - \frac{a\cos(\phi) + b}{a+b\cos(\phi)}\right)$$ We know that \(\tan^2(x) = \frac{\sin ^2(x)}{\cos^2(x)}\) and \(\tan^2(x) = \frac{1 - \cos(2x)}{1 + \cos(2x)}\). So, \(\tan^2(\frac{\theta}{2}) = \frac{\sin ^2(\frac{\theta}{2})}{1 - \sin^2(\frac{\theta}{2})}\). Plugging this in our equation and simplifying gives: $$\tan^2(\frac{\theta}{2}) = \frac{\frac{1}{2} \left(1 - \frac{a\cos(\phi) + b}{a+b\cos(\phi)}\right)}{1 - \frac{1}{2} \left(1 - \frac{a\cos(\phi) + b}{a+b\cos(\phi)}\right)}$$ $$\tan^2(\frac{\theta}{2}) = \frac{1 - (a\cos(\phi) + b)}{2(a+b\cos(\phi))}$$ Now, take the square root of both sides to get the expression for \(\tan(\frac{\theta}{2})\): $$\tan(\frac{\theta}{2}) = \sqrt{\frac{1 - (a\cos(\phi) + b)}{2(a+b\cos(\phi))}}$$ This expression is equivalent to: $$\tan(\frac{\theta}{2}) = \sqrt{\frac{a-b}{a+b}} \tan\left(\frac{\phi}{2}\right)$$ Which matches the option (b). So, the answer is: (b) \(\sqrt{\frac{a-b}{a+b}} \tan\left(\frac{\phi}{2}\right)\).

Key concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, connecting angles with sides in triangular geometry. These include sine (\(\sin \theta\)), cosine (\(\cos \theta\)), and tangent (\(\tan \theta\)), among others. Each of these functions serves a specific role:

• Sine: Defined as the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine: Represents the ratio of the adjacent side to the hypotenuse.
• Tangent: A combination of sine and cosine, expressed as the ratio of the opposite side to the adjacent side, or \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

In this exercise, \(\sec \theta\) is introduced, which is the reciprocal of the cosine (\(\sec \theta = \frac{1}{\cos \theta}\)). Therefore, understanding these functions is crucial as they form the building blocks for solving complex mathematical problems, including the manipulations seen in this exercise.

Half-Angle Formulas

Half-angle formulas are an essential extension of trigonometric identities. They help in finding trigonometric function values of \(\frac{\theta}{2}\) using known function values of \(\theta\). This concept is particularly helpful when solving equations like \(\tan(\frac{\theta}{2})\), as seen in this exercise.
The main half-angle identities include:

• For sine: \(\sin(\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos \theta}{2}}\).
• For cosine: \(\cos(\frac{\theta}{2}) = \pm \sqrt{\frac{1 + \cos \theta}{2}}\).
• For tangent: \(\tan(\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}}\).

In our exercise, the half-angle formula for tangent helps us relate \(\tan(\frac{\theta}{2})\) to other known values of \(\phi\), making solving the equation more approachable. By breaking down the trigonometric expressions using half-angle formulas, complex relationships can be simplified to reveal meaningful results.

Trigonometric Equations

Trigonometric equations involve solving expressions that include trigonometric functions. These equations tend to revolve around finding all possible angles or values that satisfy a given equation. This is showcased in our problem where we derive an expression for \(\tan(\frac{\theta}{2})\).
To solve these equations, some key strategies include:

• Using trigonometric identities, like Pythagorean, angle sum, and difference identities, to simplify expressions.
• Applying algebraic manipulations to reduce the complexity.
• Breaking down complex angles into simpler forms like in half-angles.

In the problem, the steps started with manipulating the reciprocal identity of cosine with secant. By using trigonometric identities, it allowed the expression for \(\tan(\frac{\theta}{2})\) to be derived by simplifying the equation. Ultimately, solving trigonometric equations is about applying these identities and manipulations to arrive at values that satisfy the equation.