Math  /  Trigonometry

Question8. Write each expression as a single trigonometric function. a) sin28cos35+cos28sin35\sin 28^{\circ} \cos 35^{\circ}+\cos 28^{\circ} \sin 35^{\circ} b) cos10cos7sin10sin7\cos 10^{\circ} \cos 7^{\circ}-\sin 10^{\circ} \sin 7^{\circ} d) sinπ3cosπ4cosπ3sinπ4\sin \frac{\pi}{3} \cos \frac{\pi}{4}-\cos \frac{\pi}{3} \sin \frac{\pi}{4}

Studdy Solution

STEP 1

What is this asking? We need to rewrite some expressions that look like a mess of sines and cosines into a *single*, beautiful, trigonometric function! Watch out! Remember those angle sum and difference formulas?
They're the key here!
Don't mix them up!

STEP 2

1. Solve for (a)
2. Solve for (b)
3. Solve for (d)

STEP 3

We've got sin28cos35+cos28sin35\sin 28^{\circ} \cos 35^{\circ} + \cos 28^{\circ} \sin 35^{\circ}.
This looks a *lot* like the angle sum formula for sine, which is sin(a+b)=sinacosb+cosasinb\sin(a+b) = \sin a \cos b + \cos a \sin b.

STEP 4

Let's **match things up**!
We can see that a=28a = 28^{\circ} and b=35b = 35^{\circ}.
So, we can rewrite our expression as sin(28+35)\sin(28^{\circ} + 35^{\circ}).

STEP 5

Adding those angles gives us sin(63)\sin(63^{\circ}).
Boom! A single trigonometric function!

STEP 6

Now we have cos10cos7sin10sin7\cos 10^{\circ} \cos 7^{\circ} - \sin 10^{\circ} \sin 7^{\circ}.
This time, it resembles the angle sum formula for cosine: cos(a+b)=cosacosbsinasinb\cos(a+b) = \cos a \cos b - \sin a \sin b.

STEP 7

Again, let's **match things up**!
We have a=10a = 10^{\circ} and b=7b = 7^{\circ}.
So, our expression becomes cos(10+7)\cos(10^{\circ} + 7^{\circ}).

STEP 8

Adding the angles gives us cos(17)\cos(17^{\circ}).
Another single trigonometric function!
We're on a roll!

STEP 9

This one has radians!
No problem!
We have sinπ3cosπ4cosπ3sinπ4\sin \frac{\pi}{3} \cos \frac{\pi}{4} - \cos \frac{\pi}{3} \sin \frac{\pi}{4}.
This looks like the angle difference formula for sine: sin(ab)=sinacosbcosasinb\sin(a-b) = \sin a \cos b - \cos a \sin b.

STEP 10

Matching up, we get a=π3a = \frac{\pi}{3} and b=π4b = \frac{\pi}{4}.
So, our expression is sin(π3π4)\sin(\frac{\pi}{3} - \frac{\pi}{4}).

STEP 11

Subtracting those fractions (remember to find a common denominator!), we get π3π4=4π123π12=π12. \frac{\pi}{3} - \frac{\pi}{4} = \frac{4\pi}{12} - \frac{3\pi}{12} = \frac{\pi}{12}. So, our final answer is sin(π12)\sin(\frac{\pi}{12}).
Fantastic!

STEP 12

a) sin(63)\sin(63^{\circ}) b) cos(17)\cos(17^{\circ}) d) sin(π12)\sin(\frac{\pi}{12})

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