Question

homework

1. write the sin, cos, and tan for angle a.
2. a triangle has side lengths of 5, 15, and 12. is it a right, acute, or obtuse triangle?
3. which of the following is \\(\frac{30}{34}\\)?

select all that apply.
a) \\(\sin a\\)
b) \\(\cos a\\)
c) \\(\tan a\\)
d) \\(\sin b\\)
e) \\(\cos b\\)
ab) \\(\tan b\\)

1. a 25-foot ladder comes with instructions that say you cannot place the base of the ladder closer than 5 feet to the wall. if you place the base of the 5 feet from the wall, how far up the wall will the ladder reach?
2. write the sin, cos, and tan for angle c.
3. find st.
4. write all three trigonometric ratios for the given angle below.
5. what is the value of x?

Explanation:

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<reasoning>

Classify triangle in Question 2

Using the Pythagorean Theorem knowledge point
\[

$$\begin{aligned} &a = 5, \quad b = 12, \quad c = 15\\ &a^2 + b^2 = 5^2 + 12^2 = 25 + 144 = 169\\ &c^2 = 15^2 = 225\\ &c^2 > a^2 + b^2 \implies 225 > 169 \end{aligned}$$

\]

Solve for ST in Question 6

Using the Pythagorean Theorem knowledge point
\[

$$\begin{aligned} &ST^2 + TU^2 = SU^2\\ &ST^2 + 30^2 = 50^2\\ &ST^2 + 900 = 2500\\ &ST^2 = 1600 \implies ST = 40 \end{aligned}$$

\]

Solve for x in Question 8

Using the Pythagorean Theorem knowledge point
\[

$$\begin{aligned} &x^2 = 7^2 + (5\sqrt{2})^2\\ &x^2 = 49 + 25 \cdot 2\\ &x^2 = 49 + 50 = 99\\ &x = \sqrt{99} = 3\sqrt{11} \end{aligned}$$

\]

Determine trigonometric ratios for Questions 1, 3, 5, 7

We apply the definitions of sine, cosine, and tangent:

• \(\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}\)
• \(\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}\)
• \(\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}\)

For Question 1:

• \(\sin A = \frac{16}{34} = \frac{8}{17}\)
• \(\cos A = \frac{30}{34} = \frac{15}{17}\)
• \(\tan A = \frac{16}{30} = \frac{8}{15}\)

For Question 3:

• \(\frac{30}{34}\) represents the ratio of the side adjacent to \(A\) (which is \(30\)) to the hypotenuse (\(34\)), which is \(\cos A\).
• It also represents the ratio of the side opposite to \(B\) (which is \(30\)) to the hypotenuse (\(34\)), which is \(\sin B\).
• Thus, the correct options are b and d.

For Question 5:

• \(\sin C = \frac{35}{37}\)
• \(\cos C = \frac{12}{37}\)
• \(\tan C = \frac{35}{12}\)

For Question 7:

• \(\sin(35^\circ) = \frac{36}{h}\)
• \(\cos(35^\circ) = \frac{52}{h}\)
• \(\tan(35^\circ) = \frac{36}{52} = \frac{9}{13}\)

Solve the ladder word problem in Question 4

Using the Pythagorean Theorem knowledge point
\[

$$\begin{aligned} &\text{ladder} = 25, \quad \text{base} = 5\\ &\text{height}^2 + 5^2 = 25^2\\ &\text{height}^2 + 25 = 625\\ &\text{height}^2 = 600 \implies \text{height} = \sqrt{600} = 10\sqrt{6} \approx 24.49\text{ ft} \end{aligned}$$

\]
</reasoning>

<answer>

No.	Answer
2	Obtuse
3	b) \(\cos A\), d) \(\sin B\)
4	\(10\sqrt{6}\text{ ft}\) (approx. \(24.49\text{ ft}\))
5	\(\sin C = \frac{35}{37}\), \(\cos C = \frac{12}{37}\), \(\tan C = \frac{35}{12}\)
6	\(40\)
7	\(\sin(35^\circ) = \frac{36}{h}\), \(\cos(35^\circ) = \frac{52}{h}\), \(\tan(35^\circ) = \frac{9}{13}\)
8	\(3\sqrt{11}\text{ in}\)

</answer>

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Answer:

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<reasoning>

Classify triangle in Question 2

Using the Pythagorean Theorem knowledge point
\[

$$\begin{aligned} &a = 5, \quad b = 12, \quad c = 15\\ &a^2 + b^2 = 5^2 + 12^2 = 25 + 144 = 169\\ &c^2 = 15^2 = 225\\ &c^2 > a^2 + b^2 \implies 225 > 169 \end{aligned}$$

\]

Solve for ST in Question 6

Using the Pythagorean Theorem knowledge point
\[

$$\begin{aligned} &ST^2 + TU^2 = SU^2\\ &ST^2 + 30^2 = 50^2\\ &ST^2 + 900 = 2500\\ &ST^2 = 1600 \implies ST = 40 \end{aligned}$$

\]

Solve for x in Question 8

Using the Pythagorean Theorem knowledge point
\[

$$\begin{aligned} &x^2 = 7^2 + (5\sqrt{2})^2\\ &x^2 = 49 + 25 \cdot 2\\ &x^2 = 49 + 50 = 99\\ &x = \sqrt{99} = 3\sqrt{11} \end{aligned}$$

\]

Determine trigonometric ratios for Questions 1, 3, 5, 7

We apply the definitions of sine, cosine, and tangent:

• \(\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}\)
• \(\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}\)
• \(\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}\)

For Question 1:

• \(\sin A = \frac{16}{34} = \frac{8}{17}\)
• \(\cos A = \frac{30}{34} = \frac{15}{17}\)
• \(\tan A = \frac{16}{30} = \frac{8}{15}\)

For Question 3:

• \(\frac{30}{34}\) represents the ratio of the side adjacent to \(A\) (which is \(30\)) to the hypotenuse (\(34\)), which is \(\cos A\).
• It also represents the ratio of the side opposite to \(B\) (which is \(30\)) to the hypotenuse (\(34\)), which is \(\sin B\).
• Thus, the correct options are b and d.

For Question 5:

• \(\sin C = \frac{35}{37}\)
• \(\cos C = \frac{12}{37}\)
• \(\tan C = \frac{35}{12}\)

For Question 7:

• \(\sin(35^\circ) = \frac{36}{h}\)
• \(\cos(35^\circ) = \frac{52}{h}\)
• \(\tan(35^\circ) = \frac{36}{52} = \frac{9}{13}\)

Solve the ladder word problem in Question 4

Using the Pythagorean Theorem knowledge point
\[

$$\begin{aligned} &\text{ladder} = 25, \quad \text{base} = 5\\ &\text{height}^2 + 5^2 = 25^2\\ &\text{height}^2 + 25 = 625\\ &\text{height}^2 = 600 \implies \text{height} = \sqrt{600} = 10\sqrt{6} \approx 24.49\text{ ft} \end{aligned}$$

\]
</reasoning>

<answer>

No.	Answer
2	Obtuse
3	b) \(\cos A\), d) \(\sin B\)
4	\(10\sqrt{6}\text{ ft}\) (approx. \(24.49\text{ ft}\))
5	\(\sin C = \frac{35}{37}\), \(\cos C = \frac{12}{37}\), \(\tan C = \frac{35}{12}\)
6	\(40\)
7	\(\sin(35^\circ) = \frac{36}{h}\), \(\cos(35^\circ) = \frac{52}{h}\), \(\tan(35^\circ) = \frac{9}{13}\)
8	\(3\sqrt{11}\text{ in}\)

</answer>

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