Point N is on hypotenuse BC of triangle ABC such than angle CAN is 45 degrees.
If AC=8 and AB=6, find AN.
To find the length of AN, we need to use the trigonometric ratios in a right-angled triangle.
First, let's draw triangle ABC with the given information. The triangle will have a right angle at angle C.
A
|\
| \
//\
// \
// \
B---------C
Now, let's label the points:
- A is the angle opposite to side BC (hypotenuse).
- B is the angle opposite to side AC (adjacent).
- C is the right angle.
- N is a point on BC.
Given information:
- AC = 8 units
- AB = 6 units
- Angle CAN = 45 degrees
To find AN, we can use the sine or cosine ratio.
Using the sine ratio:
sin(angle) = opposite/hypotenuse
In this case, we have angle CAN, where AN is the opposite side and AC is the hypotenuse.
sin(CAN) = AN / AC
We can rearrange this equation to solve for AN:
AN = sin(CAN) * AC
Now, let's substitute the known values:
AN = sin(45 degrees) * 8
To find the value of sin(45 degrees), you can use a trigonometric table, calculator, or you can approximate it by knowing that sin(45 degrees) = 1/sqrt(2) = 0.7071.
AN = 0.7071 * 8
AN ≈ 5.657 units
Therefore, the length of AN is approximately 5.657 units.