Use the half-angle identities to evaluate [tex]$\cos 22.5^{\circ}$[/tex] exactly.
Which half-angle formula should be used to find the exact value of [tex]$\cos 22.5^{\circ}$[/tex]? Select the correct choice below and fill in the answer boxes to complete (Do not include the degree symbol in your answers.)
A. [tex]$\cos 22.5^{\circ}=\cos \frac{\square^{\circ}}{2}=-\sqrt{\frac{1+\cos \square^{\circ}}{2}}$[/tex]
B. [tex]$\cos 22.5^{\circ}=\cos \frac{\square^{\circ}}{2}=\sqrt{\frac{1-\cos \square^{\circ}}{2}}$[/tex]
C. [tex]$\cos 22.5^{\circ}=\cos \frac{\square^{\circ}}{2}=\sqrt{\frac{1+\cos \square^{\circ}}{2}}$[/tex]
D. [tex]$\cos 22.5^{\circ}=\cos \frac{\square^{\circ}}{2}=-\sqrt{\frac{1-\cos \square^{\circ}}{2}}$[/tex]
Answer (2)
The exact value of cos 22. 5 ∘ can be found using the half-angle identity: cos 22. 5 ∘ = 2 1 + c o s 4 5 ∘ where the blank is filled with 45 . Thus, option C is the correct choice. The final expression is cos 22. 5 ∘ = 2 2 + 2 .
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The problem requires using the half-angle formula to find the exact value of cos 22. 5 ∘ .
Since 22. 5 ∘ is in the first quadrant, cos 22. 5 ∘ is positive.
The correct half-angle formula is cos 22. 5 ∘ = 2 1 + c o s 4 5 ∘ .
Therefore, the correct choice is C, and the blank should be filled with 45: 45 .
Explanation
Understanding the Problem We are asked to find the correct half-angle formula for evaluating cos 22. 5 ∘ exactly. We need to choose the correct formula and fill in the blank with the appropriate angle.
Applying the Half-Angle Formula Since 22. 5 ∘ is in the first quadrant, its cosine is positive. The half-angle formula for cosine is given by cos 2 θ = ± 2 1 + cos θ Since cos 22. 5 ∘ is positive, we take the positive square root: cos 22. 5 ∘ = 2 1 + cos θ
Finding the Correct Angle We want to find an angle θ such that 2 θ = 22. 5 ∘ . Multiplying both sides by 2, we get θ = 4 5 ∘ . Therefore, we can write cos 22. 5 ∘ = cos 2 4 5 ∘ = 2 1 + cos 4 5 ∘ Comparing this with the given options, we see that option C matches the correct form.
Final Answer Thus, the correct choice is C, and the blank should be filled with 45. cos 22. 5 ∘ = cos 2 4 5 ∘ = 2 1 + cos 4 5 ∘
Examples
Half-angle formulas are useful in fields like surveying and navigation, where precise angle calculations are essential. For instance, if a surveyor needs to determine the cosine of a small angle but only has the cosine value of its double, they can use the half-angle formula. This is also applicable in creating accurate maps or setting up satellite communication angles, ensuring precise alignment and signal strength. By understanding these trigonometric identities, professionals can perform accurate calculations even with limited data, leading to more reliable results in their respective fields.
