In a square grid, the position of $\triangle ABC$ is shown. Find $\tan B$.

Step 1: Identify Coordinates of Vertices

First, we need to determine the coordinates of the vertices of the triangle from the given square grid. Let's assume the bottom-left corner of the grid is the origin $(0{,}0)$. Counting the grid units, we find the coordinates of A, B, and C.

Step 2: Draw an Altitude from A to BC

To find $\tan B$, we need to form a right-angled triangle. We can draw an altitude from vertex A perpendicular to the base BC. Let's call the point where the altitude meets BC as D. Since BC lies on the x-axis and A is at $(3{,}4)$, the point D will be $(3{,}0)$.

Step 3: Calculate Lengths of Sides in $\triangle$ ADB

Now we have a right-angled triangle $\triangle ADB$. We can calculate the lengths of its sides. The length of AD is the difference in the y-coordinates of A and D, which is $4-0=4$. The length of BD is the difference in the x-coordinates of B and D, which is $5-3=2$.

Step 4: Calculate tan B

In the right-angled triangle $\triangle ADB$, the side opposite to angle B is AD, and the side adjacent to angle B is BD. The tangent of an angle in a right-angled triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. Substituting the calculated lengths, we get $\tan B = \frac{4}{2} = 2$.

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).

All Introduction to Trigonometry solutions →