Know and use the sine, cosine and tangentratios for acute angles in calculations involvingsides and angles of a right-angled triangle

In a right-angled triangle, the sine (sin), cosine (cos), and tangent (tan) ratios of an acute angle can be used to calculate the sides and angles. Here's how:

1. Sine Ratio (sin): This is defined as the ratio of the length of the side that is opposite the angle to the length of the longest side of the triangle (hypotenuse). So, sin(θ) = Opposite/Hypotenuse.

2. Cosine Ratio (cos): This is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. So, cos(θ) = Adjacent/Hypotenuse.

3. Tangent Ratio (tan): This is defined as the ratio of the sine of the angle to the cosine of the angle. It's also the ratio of the side opposite the angle to the side adjacent. So, tan(θ) = Opposite/Adjacent.

To use these ratios in calculations involving sides and angles of a right-angled triangle, follow these steps:

1. Identify the angle you're working with.
2. Determine which sides of the triangle are the hypotenuse, adjacent, and opposite relative to the angle.
3. Choose the appropriate ratio (sin, cos, or tan) based on what you're trying to find.
4. Set up the equation and solve for the unknown.

For example, if you know the angle and the length of the hypotenuse, and you want to find the length of the side opposite the angle, you would use the sine ratio. If you know the lengths of the opposite and adjacent sides and want to find the angle, you would use the tangent ratio and the arctan function.