Question

Task: Name the parts of the given triangle​

Answer 1

`\textsf{1.} \quad \sf \sin(N)=(LM)/(LN)\qquad\cos(N)=(MN)/(LN)\qquad\tan(N)=(LM)/(MN)`

`\textsf{2.} \quad \sf \sin(L)=(MN)/(LN)\qquad\cos(L)=(LM)/(LN)\qquad\tan(L)=(MN)/(LM)`

Explanation:

Trigonometric ratios are mathematical functions that relate the angles of a right triangle to the ratios of the lengths of its sides.

The three primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). These ratios are defined as follows:

`\boxed{\begin{minipage}{9.4 cm}\underline{Trigonometric ratios} \\\\$\sf \sin(\theta)=(O)/(H)\quad\cos(\theta)=(A)/(H)\quad\tan(\theta)=(O)/(A)$\\\\where:\\ \phantom{ww}$\bullet$ $\theta$ is the angle. \\ \phantom{ww}$\bullet$ $\sf O$ is the side opposite the angle. \\\phantom{ww}$\bullet$ $\sf A$ is the side adjacent the angle. \\\phantom{ww}$\bullet$ $\sf H$ is the hypotenuse (the side opposite the right angle). \\\end{minipage}}`

Question 1

From inspection of the given diagram, we can see that the indicated angle in right triangle LMN is ∠N.

The side opposite ∠N is LM, the side adjacent ∠N is MN, and the hypotenuse of the right triangle is LN. Therefore:

• θ = N
• O = LM
• A = MN
• H = LN

Substitute these values into the trigonometric ratios:

`\sf \sin(N)=(LM)/(LN)\qquad\cos(N)=(MN)/(LN)\qquad\tan(N)=(LM)/(MN)`

`\hrulefill`

Question 2

From inspection of the given diagram, we can see that the indicated angle in right triangle LMN is ∠L.

The side opposite ∠L is MN, the side adjacent ∠L is LM, and the hypotenuse of the right triangle is LN. Therefore:

• θ = N
• O = MN
• A = LM
• H = LN

Substitute these values into the trigonometric ratios:

`\sf \sin(L)=(MN)/(LN)\qquad\cos(L)=(LM)/(LN)\qquad\tan(L)=(MN)/(LM)`