Question

Write the ratios for sin A and cos A. Not drawn to scale sinA= 24/10, cosA = 10/26 sinA= 24/26, cosA = 10/24

Answer 1

The correct ratios are
`\( \sin A = (10)/(26) \)` and
`\( \cos A = (24)/(26) \).`(last option)

To find the ratios for sin A and cos A for a right-angled triangle, we can use the definitions of sine and cosine in relation to the sides of the triangle.

The sine of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the hypotenuse. The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.

The triangle provided in the image is labeled with side lengths: the side opposite angle A is 10 units, the side adjacent to angle A is 24 units, and the hypotenuse (opposite the right angle) is 26 units.

Using these definitions:

-
`\( \sin A = \frac{\text{opposite side}}{\text{hypotenuse}} = (10)/(26) \)`

-
`\( \cos A = \frac{\text{adjacent side}}{\text{hypotenuse}} = (24)/(26) \)`

These ratios can be simplified by dividing both the numerator and the denominator by the greatest common divisor of each pair of numbers.

Let's simplify these ratios.

The simplified ratios for sin A and cos A are:

`\( \sin A = (5)/(13) \)`

`\( \cos A = (12)/(13) \)`

Answer 2

To find the values of trigonometric functions, we can use the acronym SOHCAHTOA, which stands for:

`\begin{gathered} \sin\theta=(opposite)/(hypotenue) \\ \\ \cos\theta=(adjacent)/(hypotenuse) \\ \\ \tan\theta=(opposite)/(adjacent) \end{gathered}`

Because we are only looking for sin A and cos A, we'll use SOH and CAH. Remember that the positions are relative to angle A. So side BC would be the opposite and side AC would be the adjacent sides.

`\begin{gathered} \sin A=(24)/(26)\text{ }or\text{ }(12)/(13) \\ \\ cosA=(10)/(26)\text{ }or\text{ }(5)/(13) \end{gathered}`

The answer is the third option, sin A = 24/26, cos A = 10/26.