Question

What happens if you try to apply the law of sines to a right triangle?

Short answer

Applying the law of sines to a right triangle gives the ratio of the length of the hypotenuse to either of the other sides equals the sine of the angle adjacent to that side. This is because the sine of a 90 degrees or \(\pi/2\) radian angle is 1, leading to a simplification of the law of sines.

Step-by-step solution

Identify the Right Triangle

In a right triangle, one angle is always 90 degrees (also written as \( \pi/2 \) radians). Call this angle C and its adjacent sides a and b. The hypotenuse is c, and the remaining angle is A, with B = 90 degrees.

Applying the Law of Sines

The law of sines states that the ratio of a side to the sine of its opposite angle is constant. So, we have \( a/ \sin(A) = c/ \sin(C) \), and \( b/ \sin(B) = c/ \sin(C) \). But since B is 90 degrees or \( \pi/2 \) radians, \( \sin(B) = 1 \), which gives \( b/c = \sin(C) = a/b \).

Interpret the Result

This means that for any right triangle, the ratio of the length of the side opposite the right angle (the hypotenuse) to any other side equals the sine of the angle adjacent to that side. This is a special case of the law of sines that applies only to right triangles and comes directly from the definition of sine in a unit circle.

Key concepts

Law of Sines

The Law of Sines is a fundamental principle in trigonometry that applies to any triangle. It states that in a triangle, the ratio of the length of a side to the sine of its opposite angle is the same for all three sides. This can be expressed as:

• \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \]

where \( a, b, \) and \( c \) are the sides of the triangle and \( A, B, \) and \( C \) are the respective opposite angles. This law is especially useful for solving non-right triangles, where simple trigonometric identities might not work.

However, the Law of Sines also holds true for right triangles with specific simplifications. In a right triangle, since one of the angles is always 90 degrees, the expression simplifies because \( \sin(90^{\circ}) = 1 \). As a result, certain ratios directly equate the sides to other trigonometric identities, leading to a straightforward solution.

Right Triangle

A right triangle is a type of triangle where one of the angles is exactly 90 degrees. This distinct feature makes it unique and simplifies many trigonometric calculations. The sides of a right triangle include:

• The hypotenuse: the longest side, opposite the right angle.
• The opposite side: the side opposite to a given angle (excluding the right angle).
• The adjacent side: the side next to a given angle and adjacent to the right angle.

The right triangle serves as the foundation for basic trigonometric ratios used in mathematics, such as sine, cosine, and tangent. When using these ratios:

• The sine of an angle is the length of the opposite side divided by the hypotenuse.
• Cosine is the length of the adjacent side divided by the hypotenuse.
• Tangent is the length of the opposite side divided by the adjacent side.

In trigonometry, understanding the properties of a right triangle is crucial, as it simplifies calculations using these trigonometric ratios.

Sine Function

The sine function is a fundamental concept in trigonometry and is defined for an angle as the ratio of the length of the opposite side to the hypotenuse in a right triangle. For any angle \(A\), the sine function is represented as:

• \[ \sin(A) = \frac{\text{opposite side}}{\text{hypotenuse}} \]

The sine function is part of the basic trigonometric functions and is periodic with a cycle of 360 degrees or \( 2\pi \) radians. This means the sine function repeats its values every full rotation around a circle.

Special values of the sine function occur at common angles:

• \( \sin(0^{\circ}) = 0\)
• \( \sin(30^{\circ}) = 0.5\)
• \( \sin(45^{\circ}) = \.\text{approximately } 0.707\)
• \( \sin(60^{\circ}) = \.\text{approximately } 0.866\)
• \( \sin(90^{\circ}) = 1\)

These values are often used in mathematical calculations involving right triangles and periodic functions.

Hypotenuse

The hypotenuse is a fundamental component of a right triangle. It is always the longest side of the triangle and lies opposite the right angle. Understanding the role of the hypotenuse is crucial when applying trigonometric ratios, including the sine, cosine, and tangent functions.

In a right triangle, the hypotenuse is used to define the sine function as the ratio of the opposite side to the hypotenuse. Similarly, the cosine of an angle is the ratio of the adjacent side to the hypotenuse:

• \[ \cos(A) = \frac{\text{adjacent side}}{\text{hypotenuse}} \]

The Pythagorean theorem also relates to the hypotenuse, stating that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides:

• \[ c^2 = a^2 + b^2 \]

where \( c \) is the length of the hypotenuse, and \( a \) and \( b \) are the lengths of the other two sides. Recognizing the properties of the hypotenuse aids in solving for unknown side lengths and understanding the deeper relationships within the triangle.