Question

Trig im in the half and double angle formula part

Answer 1

Trigonometry involves the study of angles and sides in right-angled triangles, often using trigonometric ratios such as sine, cosine, and tangent. Half and double angle formulas allow for the calculation of trigonometric values for an angle's half or double based on the original angle's values.

Step-by-step explanation:

Trigonometry is primarily concerned with the study of angles and sides of right-angled triangles. In trigonometry, the relationship between the angles and sides is expressed through various ratios which are known as trigonometric functions, like the sine (sin), cosine (cos), and tangent (tan). Particularly when working with half and double angle formulas, these functions allow us to determine the trigonometric values of an angle's half or double simply by knowing the trigonometric values of the original angle.

For example, the double angle formulas for sine and cosine are:

• 
  
• sin(2A) = 2sin(A)cos(A)
• 
  
• cos(2A) = cos2(A) - sin2(A) = 2cos2(A) - 1 = 1 - 2sin2(A)

Similarly, the half angle formulas are:

• 
  
• sin(A/2) = √[(1 - cos(A))/2]
• 
  
• cos(A/2) = √[(1 + cos(A))/2]

These formulas are immensely useful in solving trigonometric equations, finding the values of unknown angles or sides, and even in calculating certain physical phenomena in Physics and Engineering.

Answer 2

sin(u/2) = 0.9333

EXPLANATION

To answer this question, we have to use the formula for the sine of the half angle,

`\sin\left((\theta)/(2)\right)=\pm\sqrt{(1-\cos\theta)/(2)}`

In this case, the cosine of the angle is given, cos u = -0.742,

`\sin\left((u)/(2)\right)=\pm\sqrt{(1-(-0.742))/(2)}=\pm\sqrt{(1+0.742)/(2)}=\pm\sqrt{(1.742)/(2)}=\pm√(0.871)\approx\pm0.9333`

How do we decide which sign to use? This is why the quadrant information is given. The angle, u, is in quadrant III, so the half angle must be either in quadrant I or in quadrant I. Angles in these two quadrants have a positive sine. Therefore, we have to use the positive result.

Hence, the result is sin(u/2) = 0.9333.