Find the exact value of the trigonometric function at the given real number.

(a) sin(5pi/3) __

(b) cos(23pi/3) __

Determine the coefficient of

(a) x 2y 3 z 2 in (x + y + z) 7

(b) x 2y 3 z 2 in (x + 2y + z)

The Correct Answer and Explanation is:

Let’s solve the problems one by one with clear steps and explanations.

1. Trigonometric Function Values

To solve these, we reduce the angles to an equivalent angle between $0$ and $2\pi$ and then use the unit circle.

(a) $\sin\left(\frac{5\pi}{3}\right)$

Step 1: $\frac{5\pi}{3}$ is in radians.

Since $2\pi = \frac{6\pi}{3}$, we see:

$$
\frac{5\pi}{3} < 2\pi \Rightarrow \text{Angle lies within one full circle}
$$

Step 2: Find reference angle.

Since $\frac{5\pi}{3}$ is in the 4th quadrant, the reference angle is:

$$
2\pi – \frac{5\pi}{3} = \frac{\pi}{3}
$$

Step 3: Use unit circle for $\sin(\pi/3) = \frac{\sqrt{3}}{2}$

Since sine is negative in the 4th quadrant:

$$
\boxed{\sin\left(\frac{5\pi}{3}\right) = -\frac{\sqrt{3}}{2}}
$$

(b) $\cos\left(\frac{23\pi}{3}\right)$

Step 1: Simplify the angle.

$$
\frac{23\pi}{3} \div 2\pi = \frac{23\pi}{3} \cdot \frac{1}{2\pi} = \frac{23}{6} \Rightarrow \text{That’s 3 full circles and } \frac{5\pi}{3}
$$

So,

$$
\frac{23\pi}{3} \equiv \frac{5\pi}{3} \mod 2\pi
$$

Step 2: Use cosine of $\frac{5\pi}{3}$

As we saw earlier, reference angle is $\frac{\pi}{3}$, cosine is positive in 4th quadrant:

$$
\cos\left(\frac{5\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}
$$

$$
\boxed{\cos\left(\frac{23\pi}{3}\right) = \frac{1}{2}}
$$

(c) $\tan\left(\frac{5\pi}{3}\right)$

Use:

$$
\tan\left(\frac{5\pi}{3}\right) = \frac{\sin\left(\frac{5\pi}{3}\right)}{\cos\left(\frac{5\pi}{3}\right)} = \frac{-\frac{\sqrt{3}}{2}}{\frac{1}{2}} = -\sqrt{3}
$$

$$
\boxed{\tan\left(\frac{5\pi}{3}\right) = -\sqrt{3}}
$$

2. Coefficient in Multinomial Expansion

We apply the Multinomial Theorem for expressions of the form $(x + y + z)^n$:

$$
(x + y + z)^n = \sum \frac{n!}{a!b!c!} x^a y^b z^c
$$

where $a + b + c = n$

(a) Coefficient of $x^2 y^3 z^2$ in $(x + y + z)^7$

We use:

$a = 2, b = 3, c = 2$
$a + b + c = 7$, so valid.

$$
\text{Coefficient} = \frac{7!}{2!3!2!} = \frac{5040}{2 \cdot 6 \cdot 2} = \frac{5040}{24} = \boxed{210}
$$

(b) Coefficient of $x^2 y^3 z^2$ in $(x + 2y + z)^7$

This is not just a multinomial coefficient. Since there is a coefficient inside the expression (i.e., 2 with $y$), we compute:

Use multinomial coefficient:

$$
\frac{7!}{2!3!2!} = 210 \quad \text{(same as above)}
$$

But account for the extra factor from $(2y)^3 = 8y^3$

So the actual coefficient is:

$$
210 \cdot 8 = \boxed{1680}
$$

Detailed Explanation (300+ Words)

In part one, we deal with trigonometric functions of angles given in radians. Since trigonometric functions are periodic, we reduce angles to a principal value within $0$ to $2\pi$. The sine, cosine, and tangent functions follow known rules in each quadrant of the unit circle. For example, $\sin\left(\frac{5\pi}{3}\right)$ lies in the 4th quadrant where sine is negative. The reference angle is $\frac{\pi}{3}$, and the sine of that is $\frac{\sqrt{3}}{2}$, so the final value is negative. Similarly, cosine in the 4th quadrant is positive, and tangent is negative. When evaluating $\cos\left(\frac{23\pi}{3}\right)$, we reduce the angle by subtracting multiples of $2\pi$ until it falls within one circle, and we again land at $\frac{5\pi}{3}$, giving the same cosine value as before.

In part two, we use the multinomial theorem, which generalizes the binomial theorem to more than two terms. The multinomial coefficient $\frac{n!}{a!b!c!}$ tells us how many times the term $x^a y^b z^c$ appears in the expansion of $(x + y + z)^n$. In problem (a), we compute the coefficient of $x^2 y^3 z^2$ in the expansion of $(x + y + z)^7$, resulting in $\frac{7!}{2!3!2!} = 210$. In problem (b), because one term has a coefficient (i.e., $2y$), we must include this extra multiplier. So, $(2y)^3 = 8y^3$, meaning the coefficient becomes $210 \cdot 8 = 1680$. This illustrates the importance of paying attention to both exponents and numerical coefficients in algebraic expansions.