Problem 50 - Algebra 2

Question

Use the definitions of trigonometric ratios in right \(\triangle A B C\) to verify each identity. \(\tan A=\frac{\sin A}{\cos A}\)

Step-by-Step Solution

Answer

The given trigonometric identity \(\tan A=\frac{\sin A}{\cos A}\) is verified using the definitions of sine, cosine and tangent in a right triangle. The substitution leads then to \(\tan A = \frac{\frac{opposite}{hypotenuse}}{\frac{adjacent}{hypotenuse}}\), which simplifies back to \(\tan A = \frac{opposite}{adjacent}\).

Step 1: Understanding the Definitions of Trigonometric Ratios

The trigonometric ratios in a right triangle are defined in terms of the triangle's sides: \(\sin A = \frac{opposite}{hypotenuse}\), \(\cos A = \frac{adjacent}{hypotenuse}\), and \(\tan A = \frac{opposite}{adjacent}\).

Step 2: Substituting the Definitions into the Given Identity

The given identity is \(\tan A=\frac{\sin A}{\cos A}\). By substituting the definitions of the trigonometric ratios, we get \(\tan A = \frac{\frac{opposite}{hypotenuse}}{\frac{adjacent}{hypotenuse}}\).

Step 3: Simplifying the Expression

The expression \(\frac{\frac{opposite}{hypotenuse}}{\frac{adjacent}{hypotenuse}}\) simplifies to \(\frac{opposite}{adjacent}\), which is the definition of the \(\tan A\) in a right triangle.

Key Concepts

Right Triangle

A right triangle is a type of triangle that has one angle equal to 90 degrees. This right angle distinguishes it from other triangles. In a right triangle, the side opposite the right angle is the longest and is called the hypotenuse. The other two sides are called the adjacent and the opposite sides, depending on the angle you are examining.
When working with right triangles, it is essential to label each side correctly as it helps in using trigonometric ratios accurately. Remember, the opposite side is opposite to the angle you are interested in, while the adjacent side is next to the angle and forms the right angle with the hypotenuse.

Trigonometric Ratios

Trigonometric ratios are ratios of the lengths of sides in a right triangle. They are constant for a given angle in the triangle and used to relate the angles to their corresponding side ratios. The primary trigonometric ratios are:

• Sine (\( \sin \): Ratio of the length of the opposite side to the hypotenuse.
• Cosine (\( \cos \): Ratio of the length of the adjacent side to the hypotenuse.
• Tangent (\( \tan \): Ratio of the length of the opposite side to the adjacent side.

These ratios are crucial for solving problems involving right triangles and are also foundational for understanding more advanced trigonometry.

Sine and Cosine

Sine and cosine are two fundamental trigonometric functions that describe the relationship between the angles and sides of a right triangle.
The sine function for an angle \( A \) is given by the formula:\( \sin A = \frac{\text{opposite}}{\text{hypotenuse}} \). It tells us how much of the hypotenuse is covered when looking from the opposite side.Cosine for the same angle \( A \) is calculated as:\( \cos A = \frac{\text{adjacent}}{\text{hypotenuse}} \). It gives the ratio of the adjacent side to the hypotenuse.
Understanding sine and cosine is critical for many applications in physics, engineering, and mathematics as they help in structuring wave functions, oscillations, and rotations.

Tangent

The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. It is expressed as:\( \tan A = \frac{\text{opposite}}{\text{adjacent}} \).
In the identity \( \tan A = \frac{\sin A}{\cos A} \), tangent is explained using sine and cosine, emphasizing how these functions interconnect. By substituting sine and cosine into the tangent ratio, you can simplify complex calculations and verify identities.
Tangent is often used in geometric calculations since it provides a direct relationship between two non-hypotenuse sides and is especially helpful when measuring inclines, slopes, and angles of elevation or depression.