Question
Express the trig ratios as fractions in simplest terms.
I
J
K
39
65
52
Answer
Attempt 2 out of 2
sin KK =
cos JJ =
sin KK and cos JJ
are equal
because start overline, I, J, end overline
IJ
is
opposite
∠KK and
adjacent to
∠JJ .
Submit Answer
I
J
K
39
65
52
Answer
Attempt 2 out of 2
sin KK =
cos JJ =
sin KK and cos JJ
are equal
because start overline, I, J, end overline
IJ
is
opposite
∠KK and
adjacent to
∠JJ .
Submit Answer
Answers
GPT-4o mini
To find the sine and cosine ratios for the angles \( \angle K \) and \( \angle J \) in a right triangle, we can assume the triangle's sides as follows based on the given lengths:
- \( IJ \) (the side opposite \( \angle K \)) = 39
- \( JK \) (the side adjacent to \( \angle J \)) = 65
- \( IK \) (the hypotenuse) = 52
However, let's clarify: the triangle's sides must adhere to the Pythagorean theorem \( a^2 + b^2 = c^2 \). Given \( JK = 65 \), which is greater than both the other sides, it appears there might have been a misunderstanding in the configuration or the labeling of the triangle sides.
In a right triangle, we typically have:
- Opposite side to angle K (let's denote this as \( a \))
- Adjacent side to angle J (let's denote this as \( b \))
- Hypotenuse (let's denote this as \( c \))
Using the right triangle properties, we can calculate:
1. **Finding the lengths of the hypotenuse (c)**:
Using Pythagorean theorem:
\[
c^2 = a^2 + b^2
\]
Here, we are missing a proper adjacent side length since \( JK \) cannot logically act as adjacent to \( J \) if it's longer than the hypotenuse.
To follow through with your ratios (assuming correct relationships for conceptual purposes):
1. **Calculate Sin and Cos**:
- **For \( \angle K \)** (sin and cos):
\[
\sin K = \frac{\text{opposite to } K}{\text{hypotenuse}} = \frac{IJ}{IK} = \frac{39}{52}
\]
- **Simplifying**:
\[
\sin K = \frac{39 \div 13}{52 \div 13} = \frac{3}{4}
\]
\[
\cos K = \frac{\text{adjacent to } K}{\text{hypotenuse}} = \frac{JK}{IK}
\]
Assuming we had a valid configuration (which we don’t based on given measurements), let’s employ the calculated ratios for a check against \( J \):
2. **For \( \angle J \)**:
\[
\sin J = \frac{\text{opposite to } J}{\text{hypotenuse}} = \frac{IJ}{IK} = \frac{39}{52} = \frac{3}{4}
\]
\[
\cos J = \frac{JK}{IK}
\]
Again needing perfect alignment in specifics for \( JK\) as adjacent, creating a mismatch in validations.
### Conclusion based on prior relations:
Thus you may need to reassess the given lengths against logical sides.
However, if treating sin \( K\) equals cos \( J\), both angles relate trigonometric sine-cosine interchange as expected for complementary angles \( K + J = 90^\circ\).
Ultimately:
\[
\sin K = \frac{3}{4} \text{, and } \cos J = \frac{3}{4} \Rightarrow \text{be cognizant of correct triangle configurations.}
\]
These relationships illustrate that they are equal based on definitions and properties derived within right triangles and their respective angles.
- \( IJ \) (the side opposite \( \angle K \)) = 39
- \( JK \) (the side adjacent to \( \angle J \)) = 65
- \( IK \) (the hypotenuse) = 52
However, let's clarify: the triangle's sides must adhere to the Pythagorean theorem \( a^2 + b^2 = c^2 \). Given \( JK = 65 \), which is greater than both the other sides, it appears there might have been a misunderstanding in the configuration or the labeling of the triangle sides.
In a right triangle, we typically have:
- Opposite side to angle K (let's denote this as \( a \))
- Adjacent side to angle J (let's denote this as \( b \))
- Hypotenuse (let's denote this as \( c \))
Using the right triangle properties, we can calculate:
1. **Finding the lengths of the hypotenuse (c)**:
Using Pythagorean theorem:
\[
c^2 = a^2 + b^2
\]
Here, we are missing a proper adjacent side length since \( JK \) cannot logically act as adjacent to \( J \) if it's longer than the hypotenuse.
To follow through with your ratios (assuming correct relationships for conceptual purposes):
1. **Calculate Sin and Cos**:
- **For \( \angle K \)** (sin and cos):
\[
\sin K = \frac{\text{opposite to } K}{\text{hypotenuse}} = \frac{IJ}{IK} = \frac{39}{52}
\]
- **Simplifying**:
\[
\sin K = \frac{39 \div 13}{52 \div 13} = \frac{3}{4}
\]
\[
\cos K = \frac{\text{adjacent to } K}{\text{hypotenuse}} = \frac{JK}{IK}
\]
Assuming we had a valid configuration (which we don’t based on given measurements), let’s employ the calculated ratios for a check against \( J \):
2. **For \( \angle J \)**:
\[
\sin J = \frac{\text{opposite to } J}{\text{hypotenuse}} = \frac{IJ}{IK} = \frac{39}{52} = \frac{3}{4}
\]
\[
\cos J = \frac{JK}{IK}
\]
Again needing perfect alignment in specifics for \( JK\) as adjacent, creating a mismatch in validations.
### Conclusion based on prior relations:
Thus you may need to reassess the given lengths against logical sides.
However, if treating sin \( K\) equals cos \( J\), both angles relate trigonometric sine-cosine interchange as expected for complementary angles \( K + J = 90^\circ\).
Ultimately:
\[
\sin K = \frac{3}{4} \text{, and } \cos J = \frac{3}{4} \Rightarrow \text{be cognizant of correct triangle configurations.}
\]
These relationships illustrate that they are equal based on definitions and properties derived within right triangles and their respective angles.