Question

1. # Write The Expression As The Sine, Cosine, Or Tangent Of An Angle. Sin 48° Cos 15° – Cos 48° Sin 15°

Trigonometry is an essential part of mathematics and it is used in a variety of fields, such as engineering, physics, astronomy and computer science. One of the most important concepts in trigonometry is the use of sine, cosine and tangent to describe angles. In this blog post, we will look at how to write the expression “sin 48° cos 15° – cos 48° sin 15°” as the sine, cosine or tangent of an angle. We will also provide some examples to help you understand this concept better.

## Trigonometric Identities

There are many trigonometric identities that can be extremely useful when trying to simplify an expression. Perhaps the most well-known identity is the Pythagorean Identity, which states that:

sin2θ + cos2θ = 1

This identity can be used to simplify expressions involving the sine and cosine of an angle. For example, if we have the expression:

sin θ – cos θ

We can use the Pythagorean Identity to simplify it as follows:

sin θ – cos θ = sin2θ – cos2θ = (1 – cos2θ) – cos2θ = 1 – 2cos2θ = 1 – 2(1 – sin2θ) = 1 – 2 + 2sin2θ = 2sin2θ

## The Sine, Cosine, and Tangent Functions

The sine, cosine, and tangent functions are defined as follows:

sin(x) = 1/2 * (cos(x-90°) – cos(x+90°))
tan(x) = 1/2 * (sin(x-90°) + sin(x+90°))
cos(x) = 1/2 * (sin(x-90°) – sin(x+90°))

These functions are periodic, with period 2π. The range of the sine and cosine functions is -1 to 1, while the range of the tangent function is -∞ to ∞.

## The Unit Circle

The unit circle is a powerful tool for understanding and working with trigonometric functions. In this section, we’ll take a closer look at what the unit circle is and how it can be used to simplify expressions involving sine, cosine, and tangent.

The unit circle is simply a circle with a radius of 1. This makes it easy to work with because all of the trigonometric functions can be expressed as ratios using the sides of a right triangle. For example, the sine of an angle can be expressed as the ratio of the length of the side opposite the angle to the length of the hypotenuse (sin = opposite/hypotenuse).

Similarly, the cosine of an angle is equal to the ratio of the length of the side adjacent to the angle to the length of the hypotenuse (cos = adjacent/hypotenuse). And finally, the tangent of an angle is equal to the ratio of the length of opposite side to that of adjacent side (tan = opposite/adjacent).

When working with angles in radians rather than degrees, these ratios remain constant regardless of what size triangle you’re working with. This means that you can always find the value of these trigonometric functions by simply measuring one side and one corner angle of a right triangle.

The unit circle also allows for easier calculation of inverse trigonometric functions. For example, if you wanted to findthe valueof sin

## Angles in Standard Position

An angle is in standard position if its vertex is at the origin, and its initial side lies along the positive x-axis. The angles shown in Figure 1 are all in standard position.

We can find the trigonometric functions of an angle in standard position by considering the right triangle formed by the angle and its initial side. For example, in Figure 2, we can see that the sine of ∠AOB is sin A = OB/OA = y/r, the cosine of ∠AOB is cos A = OA/OB = x/r, and the tangent of ∠AOB is tan A = OB/OA = y/x.

From this, we can see that to find the trigonometric functions of an angle in standard position, we simply need to find the ratio of the sides of the right triangle formed by the angle and its initial side.

## Angles in Quadrants I and II

When finding the trigonometric ratios of an angle, it is important to first identify which quadrant the angle is in. Angles in Quadrants I and II will have different values for sine, cosine, and tangent.

To find the trigonometric ratios of an angle in Quadrant I, we use the following formulas:

sin(θ) = opposite/hypotenuse

For example, if we want to find the value of sin 45°, we would use the following formula:

sin 45° = opposite/hypotenuse
= 8/17

Therefore, the value of sin 45° is approximately 0.471.

## Angles in Quadrants III and IV

In quadrants III and IV, the signs of the sine and cosine are switched. The cosine is negative in quadrant III and the sine is negative in quadrant IV. The tangent is positive in both quadrants.

## Conclusion

In this example, we demonstrated how to write an expression as the sine, cosine, or tangent of an angle. We used two angles and four trigonometric functions to calculate the answer: sin 48° cos 15° – cos 48° sin 15° = tan 33°. This shows that with a little practice, it is possible to solve complex expressions quickly and accurately using basic trigonometric principles.