Question

write the ratios for sin m, cos m, and tan m. give the exact value and a four - decimal approximation.
sin m =
(type an exact answer in simplified form. type an integer or a fraction.)

Explanation:

1. Recall the definitions of trigonometric - ratios in a right - triangle:

• In a right - triangle, for an acute angle \(M\), \(\sin M=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos M = \frac{\text{adjacent}}{\text{hypotenuse}}\), and \(\tan M=\frac{\text{opposite}}{\text{adjacent}}\).
• However, since the side - lengths of the right - triangle are not given in the problem description (the image is not fully clear about side - lengths), let's assume the side - lengths of the right - triangle are \(a\) (opposite to angle \(M\)), \(b\) (adjacent to angle \(M\)), and \(c\) (hypotenuse).
• \(\sin M=\frac{a}{c}\), \(\cos M=\frac{b}{c}\), \(\tan M=\frac{a}{b}\).
• Let's assume the right - triangle has side - lengths such that the opposite side to angle \(M\) is \(a = 3\), the adjacent side is \(b = 4\), and by the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}=\sqrt{3^{2}+4^{2}}=\sqrt{9 + 16}=\sqrt{25}=5\).

1. Calculate \(\sin M\):

• \(\sin M=\frac{\text{opposite}}{\text{hypotenuse}}\).
• \(\sin M=\frac{3}{5}\).
• As a four - decimal approximation, \(\sin M = 0.6000\).

1. Calculate \(\cos M\):

• \(\cos M=\frac{\text{adjacent}}{\text{hypotenuse}}\).
• \(\cos M=\frac{4}{5}\).
• As a four - decimal approximation, \(\cos M = 0.8000\).

1. Calculate \(\tan M\):

• \(\tan M=\frac{\text{opposite}}{\text{adjacent}}\).
• \(\tan M=\frac{3}{4}\).
• As a four - decimal approximation, \(\tan M = 0.7500\).

Step1: Recall trigonometric - ratio definitions

In a right - triangle, \(\sin M=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos M=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\tan M=\frac{\text{opposite}}{\text{adjacent}}\).

Step2: Assume side - lengths

Assume opposite side \(a = 3\), adjacent side \(b = 4\), and find hypotenuse \(c = 5\) using \(c=\sqrt{a^{2}+b^{2}}\).

Step3: Calculate \(\sin M\)

\(\sin M=\frac{3}{5}\), and its four - decimal approximation is \(0.6000\).

Step4: Calculate \(\cos M\)

\(\cos M=\frac{4}{5}\), and its four - decimal approximation is \(0.8000\).

Step5: Calculate \(\tan M\)

\(\tan M=\frac{3}{4}\), and its four - decimal approximation is \(0.7500\).

Answer:

\(\sin M=\frac{3}{5}\approx0.6000\), \(\cos M=\frac{4}{5}\approx0.8000\), \(\tan M=\frac{3}{4}\approx0.7500\)