What is the Value of \(2\tan\frac{1}{2}A\tan\frac{1}{2}B\) in Triangle ABC?

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In summary, to find the value of 2tan\frac12Atan\frac12B, we can use the given equation sin A + sin B = 2 sin C and the trigonometric identities to solve for A and B, which are both equal to \frac{\pi}{3}. Plugging in these values, we get the solution of \frac{2}{3}.
  • #1
Monoxdifly
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Suppose the angles in triangle ABC is A, B, and C. If sin A + sin B = 2 sin C, the value of \(\displaystyle 2tan\frac12Atan\frac12B\) is ...
A. \(\displaystyle \frac83\)
B. \(\displaystyle \sqrt6\)
C. \(\displaystyle \frac73\)
D. \(\displaystyle \frac23\)
E. \(\displaystyle \frac13\sqrt3\)

Since A, B, and C are the angles of triangle ABC, then C = 180° – (A + B)
sin A + sin B = 2 sin C
sin A + sin B = 2 sin(180° – (A + B))
sin A + sin B = 2 sin(A + B)
2 = \(\displaystyle \frac{sinA+sinB}{sin(A+B)}\)

\(\displaystyle 2tan\frac12Atan\frac12B\)
\(\displaystyle \frac{sinA+sinB}{sin(A+B)}×tan\frac12Atan\frac12B\)
What am I supposed to do after this?
 
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  • #2
$\sin{A} + \sin{B} = 2\sin(A+B)$

$\sin{A} + \sin{B} = 2(\sin{A}\cos{B} + \cos{A}\sin{B}) \implies \cos{B} = \cos{A} = \dfrac{1}{2} \implies A = B = \dfrac{\pi}{3}$

$2\tan^2\left(\dfrac{\pi}{6}\right) = \dfrac{2}{3}$
 
  • #3
Thank you.
 

FAQ: What is the Value of \(2\tan\frac{1}{2}A\tan\frac{1}{2}B\) in Triangle ABC?

1. What is the basic definition of trigonometry?

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is used to solve problems involving right triangles and can also be applied to other shapes and situations.

2. What are the three basic trigonometric functions?

The three basic trigonometric functions are sine, cosine, and tangent. These functions represent the ratios of the sides of a right triangle and are used to calculate the unknown sides or angles of a triangle.

3. How is trigonometry used in real life?

Trigonometry has many practical applications in fields such as engineering, physics, astronomy, and navigation. It is used to calculate distances and heights, determine angles and trajectories, and solve problems involving right triangles in real-life situations.

4. What are the key trigonometric identities?

The key trigonometric identities are sine squared plus cosine squared equals one, tangent equals sine over cosine, and the Pythagorean identities. These identities are used to simplify trigonometric expressions and equations.

5. What are the common mistakes to avoid in trigonometry?

Some common mistakes to avoid in trigonometry include mixing up the trigonometric functions and their inverses, forgetting to convert between degrees and radians, and not using the correct formula for the given problem. It is important to pay attention to units and use the appropriate trigonometric identities to avoid errors.

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