In the square grid shown, what is $\tan\angle AOB$?

Step 1: Identify Coordinates

We can set the origin $O$ at $(0{,}0)$ on the coordinate plane. By observing the grid, point $A$ is located at $(1{,}4)$ and point $B$ is located at $(4{,}1)$. Each square in the grid represents one unit.

Step 2: Calculate Slopes

The slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $m = \frac{y_2 - y_1}{x_2 - x_1}$. We calculate the slopes of lines $OA$ and $OB$.

Step 3: Apply Tangent Formula

The tangent of the angle $\theta$ between two lines with slopes $m_1$ and $m_2$ is given by the formula $\tan\theta = \frac{m_1 - m_2}{1 + m_1 m_2}$. Here, $m_1 = m_{OA}$ and $m_2 = m_{OB}$.

Step 4: Substitute and Calculate

Substitute the calculated slopes into the formula. Simplify the expression to find the value of $\tan\angle AOB$.

More from this chapter

• **EXERCISE 8.3** **1.** Express the trigonometric ratios sin A, tan A and sec A in terms of cot A.
• Cos A / 1 + sin a + 1 + sin a / cos A is equal to 2 sec a
• write all the other trigonometric ratios of angle A in terms of sec A
• Express all the trigonometric ratios in terms of Sin A
• Express the trigonometric ratio sign a sec a tan a in terms of cot a
• Prove that √(sec² A + csc² A) = tan A + cot A
• (secθ - tanθ) = (1)/(m) --- (2) Adding (1) & (2) secθ + tanθ + secθ - tanθ = (1)/(m) + (m)/(1) 2secθ = (1 + m²)/(m) secθ = (1 + m²)/(2m) (Hence proved) 2) If as
• If tanθ + secθ = m, prove that secθ = (m² + 1)/(2m).

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