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The tangent ratio is a fundamental concept in trigonometry and is used to find unknown sides and angles in right triangles. When we talk about the tangent of an angle \( \theta \), we're referring to the ratio of the opposite side to the adjacent side of a right triangle.
This can be expressed as:

• \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)

In the given exercise, the tangent ratio \( \tan \theta = 1.5 \) tells us that for every 1 unit of the adjacent side, the opposite side is 1.5 units.
This relationship is crucial in setting up our right triangle and solving for other trigonometric functions like sine and cosine. By understanding the tangent ratio, you are already establishing a solid base for working through trigonometric problems.

The Pythagorean theorem is an essential tool in trigonometry for solving problems related to right triangles. The theorem tells us that in a right triangle:

• The square of the hypotenuse is equal to the sum of the squares of the other two sides.

This can be formulated as \( c^2 = a^2 + b^2 \), where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides.
In our exercise, once we know \( \tan \theta \), we can assign the opposite side as 1.5 and the adjacent side as 1. Utilizing the Pythagorean theorem, we calculate the hypotenuse as:

• \( H = \sqrt{(1.5)^2 + 1^2} = \sqrt{3.25} \)

Calculating the hypotenuse allows us to progress to finding other trigonometric ratios like cosine and sine.

Cosine is another primary trigonometric function, which is defined as the ratio of the adjacent side to the hypotenuse in a right triangle. The mathematical expression for cosine is:

• \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)

Using our calculated values in the problem, where the adjacent side is 1 and the hypotenuse is \( \sqrt{3.25} \), the cosine of angle \( \theta \) is:

• \( \cos \theta = \frac{1}{\sqrt{3.25}} \)

Cosine, like sine and tangent, is a ratio which helps you understand the size relations within a triangle.
Knowing how each function interconnects is crucial to problem solving in trigonometry.

The sine function in trigonometry provides insights into the ratio of a right triangle's side lengths. It is defined as the ratio of the opposite side to the hypotenuse:

• \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)

In the exercise, with the opposite side as 1.5 and the hypotenuse \( \sqrt{3.25} \), the sine of angle \( \theta \) is calculated as:

• \( \sin \theta = \frac{1.5}{\sqrt{3.25}} \)

The sine function, along with cosine and tangent, form the core set of ratios used to work through right triangle problems.
By mastering these ratios, you'll find it easier to navigate trigonometric complexities.