name: ______ date: ______ per: ____1. monica put tan 42 into her calculator and got ≈ 0.9004. a. what does the 42 mean? b. what does the 0.9004 mean? c. draw a well - labeled diagram that shows what tan 42 ≈ 0.9004 means.2. your friend keeps getting confused with fractions. four of his problems are below. some he started correctly, but others are wrong. if he started correctly, show how to finish solving the equation. if he made a mistake, find and explain the error, and then fix it and solve the equation.$0.27=\frac{x}{5}$$(5)cdot0.27=\frac{x}{5}cdot(5)$$3.962=\frac{10}{y}$$(10)cdot3.962=\frac{10}{y}cdot(10)$$\frac{b}{20}=5$$(20)cdot\frac{b}{20}=5$$\frac{10}{x}=2.1$$(x)cdot\frac{10}{x}=2.1cdot(x)$

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### Problem 1
#### Part a
# Brief Explanations:
In the tangent function $	an	heta$, $	heta$ (here 42) represents an angle measure, typically in degrees (since the calculator output is a common tangent value for 42 degrees) or radians, but in this context, it's most likely degrees. So 42 is the measure of an acute angle in a right triangle (for the tangent ratio context) or the angle in the trigonometric function, measured in degrees (or radians, but degrees are common here).
# Answer:
The 42 means the measure of an angle (in degrees, most likely) for which we are calculating the tangent. Specifically, it is the measure of an acute angle in a right - triangle (for the tangent ratio interpretation) or the argument of the tangent function, with the angle typically measured in degrees.

#### Part b
# Brief Explanations:
The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side ($	an	heta=\frac{	ext{opposite}}{	ext{adjacent}}$). So $	an42\approx0.9004$ means that for an angle of 42 degrees in a right triangle, the ratio of the length of the side opposite the 42 - degree angle to the length of the side adjacent to the 42 - degree angle is approximately 0.9004.
# Answer:
The 0.9004 means the ratio of the length of the side opposite the 42 - degree angle to the length of the side adjacent to the 42 - degree angle in a right triangle (or the value of the tangent function at an angle of 42 degrees).

#### Part c
# Brief Explanations:
1. Draw a right triangle. Label one of the acute angles as 42 degrees.
2. Label the side opposite the 42 - degree angle as $opp$, the side adjacent to the 42 - degree angle as $adj$, and the hypotenuse as $hyp$.
3. Write the formula $	an(42^{\circ})=\frac{opp}{adj}\approx0.9004$. So, if we let the adjacent side length be, for example, 1 unit, the opposite side length would be approximately 0.9004 units. We can also choose other lengths for the adjacent side, say $x$, then the opposite side would be approximately $0.9004x$.
# Answer:
- Draw a right - triangle.
- Label one acute angle as $42^{\circ}$.
- Label the side opposite the $42^{\circ}$ angle as $opp$, the side adjacent as $adj$, and the hypotenuse as $hyp$.
- Include the equation $	an(42^{\circ})=\frac{opp}{adj}\approx0.9004$ near the triangle. For example, if $adj = 1$, then $opp\approx0.9004$.

### Problem 2
#### First equation: $0.27=\frac{x}{5}$ with step $(5)\cdot0.27=\frac{x}{5}\cdot(5)$
# Explanation:
## Step 1: Simplify both sides
On the left - hand side, $(5)	imes0.27 = 1.35$. On the right - hand side, $\frac{x}{5}	imes(5)=x$ (by the multiplicative inverse property, where multiplying a fraction by its denominator cancels out the denominator).
## Step 2: Solve for $x$
After simplifying, we get $x = 1.35$.
# Answer:
The solution is $x = 1.35$.

#### Second equation: $3.962=\frac{10}{y}$ with step $(10)\cdot3.962=\frac{10}{y}\cdot(10)$
# Explanation:
## Step 1: Identify the error
To solve for $y$ in the equation $3.962=\frac{10}{y}$, we should multiply both sides by $y$ (to get rid of the denominator $y$) instead of multiplying both sides by 10. The error is in the choice of the multiplier; multiplying by 10 does not isolate $y$.
## Step 2: Correct the step and solve
Multiply both sides by $y$: $y	imes3.962=\frac{10}{y}	imes y$.
Simplify both sides: $3.962y = 10$.
Then divide both sides by 3.962: $y=\frac{10}{3.962}\approx2.524$.
# Answer:
Error: Multiplied by 10 instead of $y$ to isolate $y$. Correct solution: $y=\frac{10}{3.962}\approx2.524$.

#### Third equation: $\frac{b}{20}=5$ with step $(20)\cdot\frac{b}{20}=5$
# Explanation:
## Step 1: Identify the error
To solve for $b$, we need to multiply both sides of the equation $\frac{b}{20}=5$ by 20. But the given step only multiplies the left - hand side by 20 and does not multiply the right - hand side by 20. The error is the omission of multiplying the right - hand side by 20.
## Step 2: Correct the step and solve
Multiply both sides by 20: $(20)	imes\frac{b}{20}=5	imes20$.
Simplify both sides: $b = 100$.
# Answer:
Error: Did not multiply the right - hand side by 20. Correct solution: $b = 100$.

#### Fourth equation: $\frac{10}{x}=2.1$ with step $(x)\cdot\frac{10}{x}=2.1\cdot(x)$
# Explanation:
## Step 1: Simplify both sides
On the left - hand side, $(x)	imes\frac{10}{x}=10$ (by the multiplicative inverse property, $x$ in the numerator and denominator cancels out). On the right - hand side, $2.1	imes(x)=2.1x$.
## Step 2: Solve for $x$
After simplifying, we have the equation $10 = 2.1x$. Then divide both sides by 2.1: $x=\frac{10}{2.1}\approx4.762$.
# Answer:
The solution is $x=\frac{10}{2.1}\approx4.762$.

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