Trigonometric Identities and Equations Sum and difference identities: Problem type 1: Degrees Find the exact value of cos 105^(@) by using a sum or difference formula. ]

Express as Sum of Angles: To find the exact value of cos(105°), we can express 105° as the sum or difference of angles for which we know the exact values of the cosine function. One way to do this is to express 105° as the sum of 60° and 45°, since the cosine values for these angles are known.Apply Sum Formula: Using the sum formula for cosine, which is cos(A + B) = cos(A)cos(B) - sin(A)sin(B), we can write cos(105°) as cos(60° + 45°).Substitute Known Values: Now we substitute the known values of cos(60°), cos(45°), sin(60°), and sin(45°) into the formula. These values are: cos(60°) = 1/2, cos(45°) = √2/2, sin(60°) = √3/2, sin(45°) = √2/2.Calculate Using Formula: Plugging the values into the sum formula, we get: cos(105°) = cos(60°)cos(45°) - sin(60°)sin(45°) = (1/2)(√2/2) - (√3/2)(√2/2) = √2/4 - √6/4.Combine Terms: Combining the terms, we get: cos(105°) = (√2 - √6)/4. This is the exact value of cos(105°) using the sum formula.

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Trigonometric Identities and Equations $\newline$ Sum and difference identities: Problem type $1$ : Degrees $\newline$ Find the exact value of $\newline$ $\cos 105^\circ$ by using a sum or difference formula.

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Q. Trigonometric Identities and Equations

\newline

Sum and difference identities: Problem type

1

: Degrees

\newline

Find the exact value of

\newline

\cos 105^\circ

by using a sum or difference formula.

Express as Sum of Angles: To find the exact value of $\cos(105°)$ , we can express $105°$ as the sum or difference of angles for which we know the exact values of the cosine function. One way to do this is to express $105°$ as the sum of $60°$ and $45°$ , since the cosine values for these angles are known.
Apply Sum Formula: Using the sum formula for cosine, which is $\cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B)$ , we can write $\cos(105°)$ as $\cos(60° + 45°)$ .
Substitute Known Values: Now we substitute the known values of $\cos(60°)$ , $\cos(45°)$ , $\sin(60°)$ , and $\sin(45°)$ into the formula. These values are: $\newline$ $\cos(60°) = \frac{1}{2}$ , $\newline$ $\cos(45°) = \frac{\sqrt{2}}{2}$ , $\newline$ $\sin(60°) = \frac{\sqrt{3}}{2}$ , $\newline$ $\sin(45°) = \frac{\sqrt{2}}{2}$ .
Calculate Using Formula: Plugging the values into the sum formula, we get: $\newline$ $\cos(105°) = \cos(60°)\cos(45°) - \sin(60°)\sin(45°)$ $\newline$ $= (\frac{1}{2})(\frac{\sqrt{2}}{2}) - (\frac{\sqrt{3}}{2})(\frac{\sqrt{2}}{2})$ $\newline$ $= \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}$ .
Combine Terms: Combining the terms, we get: $\newline$ $\cos(105°) = (\sqrt{2} - \sqrt{6})/4$ . $\newline$ This is the exact value of $\cos(105°)$ using the sum formula.