Math / Trigonometry

Question2. Rewrite each expression as a single trigonometric ratio, and then evaluate the ratio. a) $\frac{\tan 170^{\circ}-\tan 110^{\circ}}{1+\tan 170^{\circ} \tan 110^{\circ}}$ b) $\cos \frac{5 \pi}{12} \cos \frac{\pi}{12}+\sin \frac{5 \pi}{12} \sin \frac{\pi}{12}$

Studdy Solution

STEP 1

1. We are dealing with trigonometric identities and need to simplify expressions.
2. The expressions can be rewritten using known trigonometric identities.
3. The angles given are in degrees for part (a) and in radians for part (b).

STEP 2

1. Simplify expression (a) using the tangent subtraction identity.
2. Simplify expression (b) using the cosine addition identity.
3. Evaluate the simplified expressions.

STEP 3

For part (a), recognize the identity for the tangent of the difference of two angles:
$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$
Here, $A = 170^{\circ}$ and $B = 110^{\circ}$ .
Thus, the expression simplifies to:
$\tan(170^{\circ} - 110^{\circ})$

STEP 4

Calculate the difference:
$170^{\circ} - 110^{\circ} = 60^{\circ}$
So, the expression becomes:
$\tan 60^{\circ}$

STEP 5

Evaluate $\tan 60^{\circ}$ :
$\tan 60^{\circ} = \sqrt{3}$

STEP 6

For part (b), recognize the identity for the cosine of the sum of two angles:
$\cos(A + B) = \cos A \cos B - \sin A \sin B$
The given expression is:
$\cos \frac{5 \pi}{12} \cos \frac{\pi}{12} + \sin \frac{5 \pi}{12} \sin \frac{\pi}{12}$
This matches the identity for the cosine of the difference of two angles:
$\cos\left(\frac{5\pi}{12} - \frac{\pi}{12}\right)$

STEP 7

Calculate the difference:
$\frac{5\pi}{12} - \frac{\pi}{12} = \frac{4\pi}{12} = \frac{\pi}{3}$
So, the expression becomes:
$\cos \frac{\pi}{3}$

STEP 8

Evaluate $\cos \frac{\pi}{3}$ :
$\cos \frac{\pi}{3} = \frac{1}{2}$
The solutions are: a) $\sqrt{3}$ b) $\frac{1}{2}$

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Question2. Rewrite each expression as a single trigonometric ratio, and then evaluate the ratio. a) $\frac{\tan 170^{\circ}-\tan 110^{\circ}}{1+\tan 170^{\circ} \tan 110^{\circ}}$ b) $\cos \frac{5 \pi}{12} \cos \frac{\pi}{12}+\sin \frac{5 \pi}{12} \sin \frac{\pi}{12}$

Studdy Solution

STEP 1

1. We are dealing with trigonometric identities and need to simplify expressions.
2. The expressions can be rewritten using known trigonometric identities.
3. The angles given are in degrees for part (a) and in radians for part (b).

STEP 2

1. Simplify expression (a) using the tangent subtraction identity.
2. Simplify expression (b) using the cosine addition identity.
3. Evaluate the simplified expressions.

STEP 3

For part (a), recognize the identity for the tangent of the difference of two angles:
$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$
Here, $A = 170^{\circ}$ and $B = 110^{\circ}$ .
Thus, the expression simplifies to:
$\tan(170^{\circ} - 110^{\circ})$

STEP 4

Calculate the difference:
$170^{\circ} - 110^{\circ} = 60^{\circ}$
So, the expression becomes:
$\tan 60^{\circ}$

STEP 5

Evaluate $\tan 60^{\circ}$ :
$\tan 60^{\circ} = \sqrt{3}$

STEP 6

STEP 7

Calculate the difference:
$\frac{5\pi}{12} - \frac{\pi}{12} = \frac{4\pi}{12} = \frac{\pi}{3}$
So, the expression becomes:
$\cos \frac{\pi}{3}$

STEP 8

Evaluate $\cos \frac{\pi}{3}$ :
$\cos \frac{\pi}{3} = \frac{1}{2}$
The solutions are: a) $\sqrt{3}$ b) $\frac{1}{2}$

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Studdy Solution

STEP 1

1. We are dealing with trigonometric identities and need to simplify expressions.2. The expressions can be rewritten using known trigonometric identities.3. The angles given are in degrees for part (a) and in radians for part (b).

STEP 2

1. Simplify expression (a) using the tangent subtraction identity.2. Simplify expression (b) using the cosine addition identity.3. Evaluate the simplified expressions.

STEP 3

STEP 4

Calculate the difference:170∘−110∘=60∘170^{\circ} - 110^{\circ} = 60^{\circ}170∘−110∘=60∘So, the expression becomes:tan⁡60∘\tan 60^{\circ}tan60∘

STEP 5

Evaluate tan⁡60∘\tan 60^{\circ}tan60∘:tan⁡60∘=3\tan 60^{\circ} = \sqrt{3}tan60∘=3​

STEP 6

STEP 7

Calculate the difference:5π12−π12=4π12=π3\frac{5\pi}{12} - \frac{\pi}{12} = \frac{4\pi}{12} = \frac{\pi}{3}125π​−12π​=124π​=3π​So, the expression becomes:cos⁡π3\cos \frac{\pi}{3}cos3π​

STEP 8

Evaluate cos⁡π3\cos \frac{\pi}{3}cos3π​:cos⁡π3=12\cos \frac{\pi}{3} = \frac{1}{2}cos3π​=21​ The solutions are: a) 3\sqrt{3}3​ b) 12\frac{1}{2}21​

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1. We are dealing with trigonometric identities and need to simplify expressions.
2. The expressions can be rewritten using known trigonometric identities.
3. The angles given are in degrees for part (a) and in radians for part (b).

1. Simplify expression (a) using the tangent subtraction identity.
2. Simplify expression (b) using the cosine addition identity.
3. Evaluate the simplified expressions.

Calculate the difference:
$170^{\circ} - 110^{\circ} = 60^{\circ}$
So, the expression becomes:
$\tan 60^{\circ}$

Evaluate $\tan 60^{\circ}$ :
$\tan 60^{\circ} = \sqrt{3}$

Calculate the difference:
$\frac{5\pi}{12} - \frac{\pi}{12} = \frac{4\pi}{12} = \frac{\pi}{3}$
So, the expression becomes:
$\cos \frac{\pi}{3}$

Evaluate $\cos \frac{\pi}{3}$ :
$\cos \frac{\pi}{3} = \frac{1}{2}$
The solutions are: a) $\sqrt{3}$ b) $\frac{1}{2}$