Math Snap

STEP 1

1. We have a right triangle $\triangle KIJ$ with a right angle at $I$.
2. The side $KI$ (adjacent to $\angle K$) is 60.
3. The hypotenuse $KJ$ is 68.
4. We need to find the length of the opposite side $IJ$.
5. We need to find $\sin(K)$, $\cos(K)$, and $\tan(K)$.

STEP 2

1. Use the Pythagorean theorem to find the length of the opposite side $IJ$.
2. Calculate $\sin(K)$ using the opposite and hypotenuse.
3. Calculate $\cos(K)$ using the adjacent and hypotenuse.
4. Calculate $\tan(K)$ using the opposite and adjacent.

STEP 3

Use the Pythagorean theorem:
$$KJ^2 = KI^2 + IJ^2$$ Substitute the known values:
$$68^2 = 60^2 + IJ^2$$

STEP 4

Calculate $68^2$ and $60^2$:
$$68^2 = 4624$$ $$60^2 = 3600$$ Substitute these values back into the equation:
$$4624 = 3600 + IJ^2$$

STEP 5

Solve for $IJ^2$:
$$IJ^2 = 4624 - 3600$$ $$IJ^2 = 1024$$ Take the square root to find $IJ$:
$$IJ = \sqrt{1024}$$ $$IJ = 32$$

STEP 6

Calculate $\sin(K)$:
$$\sin(K) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{IJ}{KJ} = \frac{32}{68}$$ Simplify the fraction:
$$\sin(K) = \frac{8}{17}$$

STEP 7

Calculate $\cos(K)$:
$$\cos(K) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{KI}{KJ} = \frac{60}{68}$$ Simplify the fraction:
$$\cos(K) = \frac{15}{17}$$

Calculate $\tan(K)$:
$$\tan(K) = \frac{\text{opposite}}{\text{adjacent}} = \frac{IJ}{KI} = \frac{32}{60}$$ Simplify the fraction:
$$\tan(K) = \frac{8}{15}$$ The values are:
$$\begin{array}{l} \sin(K) = \frac{8}{17} \\ \cos(K) = \frac{15}{17} \\ \tan(K) = \frac{8}{15} \end{array}$$