Math  /  Trigonometry

QuestionFind the sine, cosine, and tangent of K\angle K.
Simplify your answers and write them as proper fractions, improper fractions, or whole numbers. sin(K)=cos(K)=tan(K)=\begin{array}{l} \sin (K)= \\ \cos (K)= \\ \tan (K)= \end{array} \square \square \square

Studdy Solution

STEP 1

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

STEP 2

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

STEP 3

Use the Pythagorean theorem:
KJ2=KI2+IJ2 KJ^2 = KI^2 + IJ^2
Substitute the known values:
682=602+IJ2 68^2 = 60^2 + IJ^2

STEP 4

Calculate 682 68^2 and 602 60^2 :
682=4624 68^2 = 4624 602=3600 60^2 = 3600
Substitute these values back into the equation:
4624=3600+IJ2 4624 = 3600 + IJ^2

STEP 5

Solve for IJ2 IJ^2 :
IJ2=46243600 IJ^2 = 4624 - 3600 IJ2=1024 IJ^2 = 1024
Take the square root to find IJ IJ :
IJ=1024 IJ = \sqrt{1024} IJ=32 IJ = 32

STEP 6

Calculate sin(K)\sin(K):
sin(K)=oppositehypotenuse=IJKJ=3268 \sin(K) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{IJ}{KJ} = \frac{32}{68}
Simplify the fraction:
sin(K)=817 \sin(K) = \frac{8}{17}

STEP 7

Calculate cos(K)\cos(K):
cos(K)=adjacenthypotenuse=KIKJ=6068 \cos(K) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{KI}{KJ} = \frac{60}{68}
Simplify the fraction:
cos(K)=1517 \cos(K) = \frac{15}{17}

STEP 8

Calculate tan(K)\tan(K):
tan(K)=oppositeadjacent=IJKI=3260 \tan(K) = \frac{\text{opposite}}{\text{adjacent}} = \frac{IJ}{KI} = \frac{32}{60}
Simplify the fraction:
tan(K)=815 \tan(K) = \frac{8}{15}
The values are:
sin(K)=817cos(K)=1517tan(K)=815\begin{array}{l} \sin(K) = \frac{8}{17} \\ \cos(K) = \frac{15}{17} \\ \tan(K) = \frac{8}{15} \end{array}

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