If $\sin \frac{A}{2}=\frac{1}{2}$, find the value of $\sin A, \cos A$ and $\tan A$.
Answer (1)
Find A /2 from sin ( A /2 ) = 1/2 , which gives A /2 = π /6 .
Calculate A = 2 ∗ A /2 = π /3 .
Determine sin A = sin ( π /3 ) = 2 3 .
Determine cos A = cos ( π /3 ) = 2 1 .
Determine tan A = c o s A s i n A = 3 .
The final answer is: sin A = 2 3 , cos A = 2 1 , tan A = 3
Explanation
Problem Analysis We are given that sin 2 A = 2 1 . Our goal is to find the values of sin A , cos A , and tan A .
Finding A/2 First, we need to find the value of angle 2 A whose sine is 2 1 . We know that sin 3 0 ∘ = sin 6 π = 2 1 . Therefore, 2 A = 6 π radians or 3 0 ∘ .
Finding A Now, we can find the value of A by multiplying 2 A by 2: A = 2 × 2 A = 2 × 6 π = 3 π So, A = 3 π radians or 6 0 ∘ .
Finding sin A Next, we find sin A . Since A = 3 π , we have: sin A = sin 3 π = 2 3 ≈ 0.866
Finding cos A Now, we find cos A . Since A = 3 π , we have: cos A = cos 3 π = 2 1 = 0.5
Finding tan A Finally, we find tan A . We know that tan A = c o s A s i n A , so: tan A = cos A sin A = 2 1 2 3 = 3 ≈ 1.732
Final Answer Therefore, the values are: sin A = 2 3 cos A = 2 1 tan A = 3
Examples
Understanding trigonometric functions like sine, cosine, and tangent is crucial in various fields. For instance, in physics, when analyzing projectile motion, you need to calculate the range, maximum height, and time of flight. These calculations heavily rely on trigonometric functions to resolve the initial velocity into horizontal and vertical components. Similarly, in engineering, when designing structures or bridges, engineers use trigonometric functions to calculate angles and forces, ensuring the stability and safety of the construction.
