To solve for the trigonometric ratios \(\sin M\), \(\cos M\), and \(\tan M\) in the given right triangle, we need to identify the sides relative to angle \(M\). Given: - \(MK = 4\) (opposite side to angle \(M\)) - \(ML = 10\) (hypotenuse) - \(KL = 2\sqrt{21}\) (adjacent side to angle \(M\)) The trigonometric ratios are defined as follows: - \(\sin M = \frac{\text{opposite}}{\text{hypotenuse}}\) - \(\cos M = \frac{\text{adjacent}}{\text{hypotenuse}}\) - \(\tan M = \frac{\text{opposite}}{\text{adjacent}}\) Now, let's calculate each ratio: 1. \(\sin M\): \[ \sin M = \frac{MK}{ML} = \frac{4}{10} = \frac{2}{5} \] 2. \(\cos M\): \[ \cos M = \frac{KL}{ML} = \frac{2\sqrt{21}}{10} = \frac{\sqrt{21}}{5} \] 3. \(\tan M\): \[ \tan M = \frac{MK}{KL} = \frac{4}{2\sqrt{21}} = \frac{2}{\sqrt{21}} = \frac{2\sqrt{21}}{21} \] So, the ratios are: - \(\sin M = \frac{2}{5}\) - \(\cos M = \frac{\sqrt{21}}{5}\) - \(\tan M = \frac{2\sqrt{21}}{21}\) For the specific question in the image, the answer for \(\sin M\) is: \[ \sin M = \frac{2}{5} \]

How can trigonometric ratios like sine, cosine, and tangent be used to find missing sides or angles in right-angled triangles?

1. With respect to a given angle, what is the ratio of the hypotenuse of a right triangle to the opposite side?*A. sineB. cosineC. secantD. cosecant

Find sinLsin𝐿 , cosLcos𝐿 , tanLtan𝐿 , sinMsin𝑀 , cosMcos𝑀 , and tanMtan𝑀 when  ℓ=12ℓ=12 ,  m=123⎯⎯√𝑚=123 , and  n=24𝑛=24 . Match each ratio to the corresponding trigonometric expression.

To solve for side \( AB \) in the given right triangle, we can use the sine function, which relates the opposite side to the hypotenuse. Given: - The angle \( \angle BCA = 20^\circ \). - The side opposite the angle \( \angle BCA \) is \( BC = 7 \). - The hypotenuse is \( AB \). We need to find \( AB \). Step 1: Write the sine function for the given angle. \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \] Step 2: Substitute the given values into the sine function. \[ \sin(20^\circ) = \frac{7}{AB} \] Step 3: Solve for \( AB \). \[ AB = \frac{7}{\sin(20^\circ)} \] Step 4: Calculate the value using a calculator. \[ \sin(20^\circ) \approx 0.3420\] \[ AB = \frac{7}{0.3420} \approx 20.47\] Therefore, the length of side \( AB \) is approximately \( 20.47 \) (rounded to the nearest hundredth).

ith respect to a given angle, what is the ratio of the hypotenuse of a right triangle to the opposite side?