Mathematics: In ΔHIJ, the measure of ∠J=90°, JI = 4, HJ = 3, and IH = 5. What ratio represents the tangent of ∠I?

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In ΔHIJ, the measure of ∠J=90°, JI = 4, HJ = 3, and

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22.01.2022

The ratio $\frac{3}{4}$ represents the tangent of ∠I

Step-by-step explanation:

Let us revise the trigonometry ratio

sin Ф = $\frac{opposite}{hypotenuse}$ cos Ф = $\frac{adjacent}{hypotenuse}$ tan Ф = $\frac{opposite}{adjacent}$

In Δ HIJ

∵ m∠J = 90°

- Hypotenuse is the side which opposite to the right angle

∴ HI is the hypotenuse

∵ HJ = 3 units

∵ IH = 5 units

- Let us use Pythagoras Theorem to find HJ

∵ (HJ)² + (IJ)² = (IH)²

∴ 3² + (IJ)² = 5²

∴ 9 + (IJ)² = 25

- Subtract 9 from both sides

∴ (IJ)² = 16

- take √ for both sides

∴ IJ = 4 units

To find the tangent of ∠I find the opposite and adjacent sides to it

∵ HJ is opposite to ∠I

∵ IJ is adjacent to ∠I

- use the rule of tan above

∴ tan(∠I) = $\frac{HJ}{IJ}$

∴ tan(∠I) = $\frac{3}{4}$

The ratio $\frac{3}{4}$ represents the tangent of ∠I

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