Math  /  Trigonometry

QuestionConsider the following. t=5π3t=\frac{5 \pi}{3} (a) Find the reference number tˉ\bar{t} for the value of tt. tˉ=π3\bar{t}=\frac{\pi}{3} (b) Find the terminal point determined by tt. (x,y)=()(x, y)=(\square) Need Help? Read It

Studdy Solution

STEP 1

What is this asking? Given an angle t=5π3t = \frac{5\pi}{3}, we need to find its reference angle and the coordinates of the point where the angle's terminal side intersects the unit circle. Watch out! Remember, the reference angle is always positive and less than π2\frac{\pi}{2}.
Also, the terminal point on the unit circle has coordinates (cos(t),sin(t))(\cos(t), \sin(t)).

STEP 2

1. Find the Reference Angle
2. Find the Terminal Point

STEP 3

Alright, so we're given t=5π3t = \frac{5\pi}{3}.
We want to find the **reference angle**, which is the smallest angle between the terminal side of our angle and the x-axis.
Think of it as the "closest" angle to the x-axis!

STEP 4

Since 5π3\frac{5\pi}{3} is in the fourth quadrant (because it's between 3π2\frac{3\pi}{2} and 2π2\pi), we can find the reference angle by subtracting tt from 2π2\pi.

STEP 5

Let's **calculate** that: 2π5π3=6π35π3=6π5π3=π32\pi - \frac{5\pi}{3} = \frac{6\pi}{3} - \frac{5\pi}{3} = \frac{6\pi - 5\pi}{3} = \frac{\pi}{3}

STEP 6

So, our **reference angle** tˉ\bar{t} is π3\frac{\pi}{3}!

STEP 7

Now, to find the **terminal point**, we need to find the x and y coordinates on the unit circle that correspond to our angle t=5π3t = \frac{5\pi}{3}.
Remember, these coordinates are given by (cos(t),sin(t))(\cos(t), \sin(t)).

STEP 8

Let's **calculate** cos(5π3)\cos(\frac{5\pi}{3}): Since 5π3\frac{5\pi}{3} is in the fourth quadrant, cosine is positive, and its value is the same as the cosine of the reference angle. cos(5π3)=cos(π3)=12\cos(\frac{5\pi}{3}) = \cos(\frac{\pi}{3}) = \frac{1}{2}

STEP 9

Now let's **calculate** sin(5π3)\sin(\frac{5\pi}{3}): Since 5π3\frac{5\pi}{3} is in the fourth quadrant, sine is negative, and its absolute value is the same as the sine of the reference angle. sin(5π3)=sin(π3)=32\sin(\frac{5\pi}{3}) = -\sin(\frac{\pi}{3}) = -\frac{\sqrt{3}}{2}

STEP 10

Therefore, the **terminal point** is (12,32)(\frac{1}{2}, -\frac{\sqrt{3}}{2}).

STEP 11

The reference angle tˉ\bar{t} is π3\frac{\pi}{3}, and the terminal point determined by tt is (12,32)(\frac{1}{2}, -\frac{\sqrt{3}}{2}).

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