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Chapter 10: Problem 2

In questions 1-11 \(\Delta \mathrm{ABC}\) has a right angle at \(\mathrm{C}\). Calculate \(\sin A\) given \(\mathrm{AC}=10 \mathrm{~cm}\) and \(\mathrm{AB}=14 \mathrm{~cm}\)

Short Answer

Expert verified
Answer: $\frac{5}{7}$

Step by step solution

01

Use the Pythagorean theorem to find the length of side \(BC\)

In a right triangle \(\Delta ABC\) with a right angle at \(C\), the Pythagorean theorem states that \(AC^2 + BC^2 = AB^2\). We are given that \(AC = 10\) cm and \(AB = 14\) cm. We can plug these values into the equation and solve for \(BC\). \(10^2 + BC^2 = 14^2\) \(100 + BC^2 = 196\) \(BC^2 = 96\) \(BC = \sqrt{96}\) cm
02

Calculate \(\sin A\) using the triangle's side lengths

In a right triangle, \(\sin A\) is defined as the ratio of the length of the opposite side (\(AC\)) to the length of the hypotenuse (\(AB\)). In this triangle, the opposite side is \(AC\) with a length of \(10\) cm and the hypotenuse is \(AB\) with a length of \(14\) cm. Thus, we can calculate \(\sin A\) as follows: \(\sin A = \frac{AC}{AB} = \frac{10}{14}\)
03

Simplify the result

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is \(2\). So we have: \(\sin A = \frac{10}{14} = \frac{5}{7}\) Thus, \(\sin A\) in triangle \(\Delta ABC\) is \(\frac{5}{7}\).

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Most popular questions from this chapter

In questions \(1-10\) solve \(\triangle \mathrm{ABC}\) given \(\mathrm{BC}=17 \mathrm{~cm}, \mathrm{AC}=27 \mathrm{~cm}, C=45^{\circ}\)

In questions 1-11 \(\Delta \mathrm{ABC}\) has a right angle at \(\mathrm{C}\). Calculate \(\mathrm{AB}\) given \(\mathrm{AC}=12 \mathrm{~cm}\) and \(A=57^{\circ}\).

Convert the following angles to degree/ minute/second format: (a) \(7.3614^{\circ}\) (b) \(10.0932^{\circ}\) (c) \(14.9610^{\circ}\)

In questions \(1-10\) solve \(\triangle \mathrm{ABC}\) given \(\mathrm{BC}=36 \mathrm{~cm}, \mathrm{AC}=92 \mathrm{~cm}, C=51^{\circ}\)

In questions 1-11 \(\Delta \mathrm{ABC}\) has a right angle at \(\mathrm{C}\). Calculate \(A\) given \(\mathrm{AC}=14 \mathrm{~cm}\) and \(\mathrm{BC}=9 \mathrm{~cm}\)
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