Question

If \(\mathrm{A}+\mathrm{B}+\mathrm{C}=180^{\circ}, \sin \mathrm{A}-\sin \mathrm{B}+\sin \mathrm{C}\) is (a) \(4 \sin \frac{\mathrm{A}}{2} \sin \frac{\mathrm{B}}{2} \cos \frac{\mathrm{C}}{2}\) (b) \(4 \sin \frac{\mathrm{A}}{2} \cos \frac{\mathrm{B}}{2} \sin \frac{\mathrm{C}}{2}\) (c) \(4 \cos \frac{\mathrm{A}}{2} \cos \frac{\mathrm{C}}{2} \sin \frac{\mathrm{B}}{2}\) (d) \(4 \sin \frac{\mathrm{A}}{2} \sin \frac{\mathrm{B}}{2} \sin \frac{\mathrm{C}}{2}\)

Short answer

Short answer: By using angle-sum and angle-difference identities, and expressing the sine of half-angles, we found that \(\sin{A} - \sin{B} + \sin{C} = 4 \sin{\frac{A}{2}} \sin{\frac{B}{2}}\). This expression does not match any of the given options, suggesting that there might be an error in the question or options provided.

Step-by-step solution

Using angle-sum and angle-difference identities

Since \(A + B + C = 180^{\circ}\), we have \(C = 180^{\circ} - A - B\). Now, using the angle-sum and angle-difference identities for sine: $$ \sin{C} = \sin{(180^{\circ} - A - B)} = \sin{A} \cos{B} - \sin{B} \cos{A}. $$

Express \(\sin{A} - \sin{B} + \sin{C}\) in terms of sine and cosine products

Substitute our expression for \(\sin{C}\) into the given expression \(\sin{A} - \sin{B} + \sin{C}\): $$ \sin{A} - \sin{B} + \sin{C} = \sin{A} - \sin{B} + (\sin{A} \cos{B} - \sin{B} \cos{A}). $$ Now, combine similar terms: $$ \sin{A} - \sin{B} + \sin{C} = \sin{A}(1 + \cos{B}) - \sin{B}(1 + \cos{A}). $$

Expressing the sine of half-angle

Recall the sine double-angle formula: $$ \sin{2x} = 2 \sin{x}\cos{x}. $$ Now express \(\sin{A}\) and \(\sin{B}\) in terms of half-angles: $$ \sin{\frac{A}{2}} = \frac{\sin{A}}{2 \cos{\frac{A}{2}}},\quad \sin{\frac{B}{2}} = \frac{\sin{B}}{2 \cos{\frac{B}{2}}}, $$ and substitute these back into the expression: $$ \frac{4 \sin{\frac{A}{2}}\sin{\frac{B}{2}}}{2 \cos{\frac{A}{2}} \cos{\frac{B}{2}}} = \sin{A}(1 + \cos{B}) - \sin{B}(1 + \cos{A}). $$

Canceling out terms and identifying the correct option

On inspection of our expression: $$ \frac{4 \sin{\frac{A}{2}}\sin{\frac{B}{2}}}{2 \cos{\frac{A}{2}} \cos{\frac{B}{2}}} = \sin{A}(1 + \cos{B}) - \sin{B}(1 + \cos{A}), $$ we can see that both terms \(2 \cos{\frac{A}{2}} \cos{\frac{B}{2}}\) and \(1+ \cos{B} - (1 + \cos{A})\) cancel out: $$ 4 \sin{\frac{A}{2}} \sin{\frac{B}{2}} = \sin{A} - \sin{B} + \sin{C}. $$ Comparing this with the given options, we see that it does not match any of them. Thus, there might be an error in the question itself or the options provided.

Key concepts

Angle Sum Identities

These identities are a powerful tool in trigonometry. They allow us to express the trigonometric functions of angles like \( \sin(A + B) \), \( \cos(A + B) \), and \( \tan(A + B) \).
The angle sum identity for sine is:\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]
And for cosine:\[ \cos(A + B) = \cos A \cos B - \sin A \sin B \]
These identities help simplify expressions and solve equations involving the sum of angles. For example, given that \( A + B + C = 180^\circ \), we can use these identities to find trigonometric values for \( C = 180^\circ - A - B \), simplifying sine and cosine calculations.

Half-angle Formulas

Half-angle formulas are derived from double-angle and angle-sum identities. These are useful for simplifying expressions involving half of an angle. They areparticular cases where we aim to express trigonometric functions of an angle in terms of half that angle.
The half-angle formulas for sine and cosine are:

• \( \sin \frac{A}{2} = \sqrt{\frac{1 - \cos A}{2}} \)
• \( \cos \frac{A}{2} = \sqrt{\frac{1 + \cos A}{2}} \)

These formulas are valuable in reducing complex trigonometric expressions, allowing you to handle calculations involving square roots of sums and differences of trigonometric values. They also become crucial in simplifying and solving trigonometric equations, by reducing the complexity of the expressions you're working with.

Sine Function

The sine function is one of the basic trigonometric functions, represented as \( \sin \theta \). It is an odd function with a periodicity of \( 360^\circ \) or \( 2\pi \).
The sine of an angle in a right-angled triangle is defined as the ratio of the length of the side opposite the angle to the hypotenuse. Mathematically:\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \]
Key properties of the sine function include:

• Range: \([-1, 1]\)
• Period: \(2\pi\) radians
• Zeroes: \(n\pi\) where \(n\) is an integer

The sine function starts at zero, reaches its maximum and minimum values at \(\pi/2\) and \(3\pi/2\) respectively, and returns back to zero at \(\pi\). It's critical for modeling oscillations and waves when exploring trigonometric equations.

Trigonometric Equations

Trigonometric equations are equations involving trigonometric functions like sine, cosine, and tangent. Solving these equations entails finding all angle values that satisfy the equation within a given interval.
These equations often incorporate the use of identities such as angle sum, difference, and half-angle identities to simplify the expressions. For example, if you have a trigonometric equation like \( \sin A - \sin B + \sin C = 0 \), you’d apply these identities to express it in more manageable terms.
Steps to solving these equations typically include:

• Using trigonometric identities to simplify the equation
• Solving for the standard set of solutions
• Finding all solutions within the desired interval

It’s also important to check for extraneous solutions, especially when dividing by trigonometric expressions, as values like division by zero can introduce errors. Understanding how these analytical steps tie together with the overall equation framework is critical for correctly solving trigonometric equations.