Question

What is the exact value of sin(theta + beta)?

Answer 1

Answer:
`(-3√(13)+4√(3))/(20)`

This is the single fraction of -3*sqrt(13)+4*sqrt(3) up top all over 20.

sqrt = square root

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Step-by-step explanation:

Angle theta is between pi and 3pi/2. This places the angle in quadrant Q3 where both cosine and sine are negative

Use the pythagorean trig identity to get the following:

`\sin^2 \theta + \cos^2 \theta = 1\\\\\sin^2 \theta + \left(-(√(3))/(4)\right)^2 = 1\\\\\sin^2 \theta + (3)/(16) = 1\\\\\sin^2 \theta = 1 - (3)/(16)\\\\\sin^2 \theta = (16)/(16) - (3)/(16)\\\\\sin^2 \theta = (16-3)/(16)\\\\\sin^2 \theta = (13)/(16)\\\\\sin \theta = -\sqrt{(13)/(16)} \ \text{ ... sine is negative in Q3}\\\\\sin \theta = -(√(13))/(√(16))\\\\\sin \theta = -(√(13))/(4)\\\\`

Angle beta is in Q1 where sine and cosine are positive.

Draw a right triangle with legs 3 and 4. The hypotenuse is 5 through the pythagorean theorem. In other words, we have a 3-4-5 right triangle.

Since
`\tan \beta = (3)/(4)`, this means
`\sin \beta = (3)/(5) \ \text{ and } \ \cos \beta = (4)/(5)`

Use these ideas:

• sin = opposite/hypotenuse
• cos = adjacent/hypotenuse
• tan = opposite/adjacent

In this case we have: opposite = 3, adjacent = 4, hypotenuse = 5.

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To recap:

`\cos \theta = -(√(3))/(4)\\\\\sin \theta = -(√(13))/(4)\\\\\cos \beta = (3)/(5)\\\\\sin \beta = (4)/(5)\\\\`

They lead to this

`\sin\left(\theta + \beta\right) = \sin \theta * \cos \beta - \cos \theta * \sin \beta\\\\\sin\left(\theta + \beta\right) = -(√(13))/(4) * (3)/(5) - \left(-(√(3))/(4)\right) * (4)/(5)\\\\\sin\left(\theta + \beta\right) = -(3√(13))/(20)+(4√(3))/(20)\\\\\sin\left(\theta + \beta\right) = (-3√(13)+4√(3))/(20)\\\\`