Questions: Find the exact value of the trigonometric expression without the use of a calculator. tan(60°+45°) Choose the expression that is equivalent to tan(60°+45°) tan(60°+45°) = (tan(60°) + tan(45°)) / (1 - tan(60°) tan(45°)) A. tan(60°+45°) = ((D) - ) / (1 + (D)()) B. tan(60°+45°) = (1 - (D)(D)) / ((D) + C) C. tan(60°+45°) = (1 + (D)(D)) / ((D) - (D)) D. tan(60°+45°) = ((D) + (D)) / (1 - (D)(D))

Find the exact value of the trigonometric expression without the use of a calculator.

tan(60°+45°)

Choose the expression that is equivalent to tan(60°+45°)

tan(60°+45°) = (tan(60°) + tan(45°)) / (1 - tan(60°) tan(45°))

A. tan(60°+45°) = ((D) - ) / (1 + (D)())
B. tan(60°+45°) = (1 - (D)(D)) / ((D) + C)
C. tan(60°+45°) = (1 + (D)(D)) / ((D) - (D))
D. tan(60°+45°) = ((D) + (D)) / (1 - (D)(D))

Transcript text: Find the exact value of the trigonometric expression without the use of a calculator. \[ \tan \left(60^{\circ}+45^{\circ}\right) \] Choose the expression that is equivalent to $\tan \left(60^{\circ}+45^{\circ}\right)$ \[ \tan \left(60^{\circ}+45^{\circ}\right)=\frac{\tan \left(60^{\circ}\right)+\tan \left(45^{\circ}\right)}{1-\tan \left(60^{\circ}\right) \tan \left(45^{\circ}\right)} \] A. $\tan \left(60^{\circ}+45^{\circ}\right)=\frac{(D)-\square}{1+(D)(\square)}$ B. $\tan \left(60^{\circ}+45^{\circ}\right)=\frac{1-(D)(D)}{(D)+C)}$ C. $\tan \left(60^{\circ}+45^{\circ}\right)=\frac{1+(D)(D)}{(D)-(D)}$ D. $\tan \left(60^{\circ}+45^{\circ}\right)=\frac{(D)+(D)}{1-(D)(D)}$

Solution

Solution Steps

To find the exact value of $\tan(60^\circ + 45^\circ)$, we can use the tangent addition formula:

\[ \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)} \]

For $a = 60^\circ$ and $b = 45^\circ$, we know $\tan(60^\circ) = \sqrt{3}$ and $\tan(45^\circ) = 1$. Substitute these values into the formula to find the exact value.

Step 1: Calculate $\tan(60^\circ)$ and $\tan(45^\circ)$

We know the values of the tangent functions: \[ \tan(60^\circ) = \sqrt{3} \approx 1.7321 \] \[ \tan(45^\circ) = 1 \]

Step 2: Apply the Tangent Addition Formula

Using the tangent addition formula: \[ \tan(60^\circ + 45^\circ) = \frac{\tan(60^\circ) + \tan(45^\circ)}{1 - \tan(60^\circ) \tan(45^\circ)} \] Substituting the known values: \[ \tan(60^\circ + 45^\circ) = \frac{\sqrt{3} + 1}{1 - \sqrt{3} \cdot 1} \]

Step 3: Simplify the Expression

Calculating the numerator and denominator: \[ \tan(60^\circ + 45^\circ) = \frac{\sqrt{3} + 1}{1 - \sqrt{3}} \approx \frac{1.7321 + 1}{1 - 1.7321} = \frac{2.7321}{-0.7321} \approx -3.7321 \]