Chapter EP, Problem 14.4.6EP

To determine

To find: the exact value of expression using half angle formula.

Answer to Problem 14.4.6EP

  ${\text{cos75}}^{\text{0}}\text{=0}\text{.2588}$ .

Explanation of Solution

Given:

  ${\text{cos75}}^0$ .

Concept used:

According to the half angle formula the given angle and $\frac{\text{θ}}{\text{2}}$ is equal.

Trigonometric half angled formula:

${\text{cos2θ=2cos}}^{\text{2}}\text{θ-1}$ and $\text{cos}\frac{\text{θ}}{\text{2}}\text{=±}\sqrt{\frac{\text{cosθ+1}}{\text{2}}}$ .

${\text{cos2θ=1-2sin}}^{\text{2}}\text{θ}$ and $\text{sin}\frac{\text{θ}}{2}\text{=±}\sqrt{\frac{\text{1-cosθ}}{\text{2}}}$ .

Calculation:

  ${\text{cos75}}^0$ .

According to the half angle formula the given angle and $\frac{\text{θ}}{\text{2}}$ is equal.

Mathematically,

  $\frac{\text{θ}}{\text{2}}{\text{=75}}^{\text{0}}$ .

  ${\text{θ=150}}^0$ .

According trigonometric half angled formula:

  $\text{cos}\frac{\text{θ}}{\text{2}}\text{=±}\sqrt{\frac{\text{cosθ+1}}{\text{2}}}$

  ${\text{cos75}}^0\text{=±}\sqrt{\frac{\text{cos150+1}}{\text{2}}}$ .

From the above unit circle the value of the angle can be deduced.

  ${\text{cos75}}^{\text{0}}\text{=±}\sqrt{\frac{\frac{\text{-}\sqrt{\text{3}}}{\text{2}}\text{+1}}{\text{2}}}$ .

  ${\text{cos75}}^{\text{0}}\text{=±}\sqrt{\frac{\text{-}\sqrt{\text{3}}+2}{4}}$ .

  ${\text{cos75}}^{\text{0}}\text{=±0}\text{.2588}$ .

Since, the angle ${75}^0$ which lies in first quadrant and in first quadrant all trigonometric ratios are positive.

Hence, ${\text{cos75}}^{\text{0}}\text{=0}\text{.2588}$ .