Problem 12 - Algebra 2 and Trigonometry

Question

Use an isosceles right triangle with legs of length 3 to find the exact values of \(\sin 45^{\circ},\) \(\cos 45^{\circ},\) and \(\tan 45^{\circ} .\)

Step-by-Step Solution

Answer

\(\sin 45^{\circ} = \cos 45^{\circ} = \frac{\sqrt{2}}{2}\), \(\tan 45^{\circ} = 1\).

Step 1: Draw the triangle

Visualize an isosceles right triangle where the two legs are of equal length. In this case, both legs are 3 units long.

Step 2: Identify the angles

Since it's an isosceles right triangle, the two non-right angles are equal. Each of these angles is \(45^{\circ}\) because the sum of angles in a triangle is \(180^{\circ}\).

Step 3: Use the Pythagorean theorem

Use the Pythagorean theorem to find the hypotenuse: \(a^2 + b^2 = c^2\). Since both legs are 3, \(3^2 + 3^2 = c^2\), giving us \(c = \sqrt{18} = 3\sqrt{2}\).

Step 4: Calculate \(\sin 45^{\circ}\)

\(\sin \theta\) is defined as the ratio of the opposite side to the hypotenuse. Thus, \(\sin 45^{\circ} = \frac{3}{3\sqrt{2}} = \frac{\sqrt{2}}{2}\) after rationalizing the denominator.

Step 5: Calculate \(\cos 45^{\circ}\)

\(\cos \theta\) is the ratio of the adjacent side to the hypotenuse. Hence, \(\cos 45^{\circ} = \frac{3}{3\sqrt{2}} = \frac{\sqrt{2}}{2}\) after rationalization.

Step 6: Calculate \(\tan 45^{\circ}\)

\(\tan \theta\) is defined as the ratio of the opposite side to the adjacent side. For this triangle, \(\tan 45^{\circ} = \frac{3}{3} = 1\).

Key Concepts

Isosceles Right Triangle

An isosceles right triangle is a special type of right triangle where two of its sides, referred to as 'legs,' are of equal length. The defining characteristic of this triangle is that it not only has a right angle, which measures \(90^{\circ}\), but also two other angles that are equal. These angles are \(45^{\circ}\) each. This formation gives the triangle its name: 'isosceles' for the two sides of equal length and 'right' because of the right angle.

When working with these triangles, knowing that the legs are equal allows us to use simple trigonometric ratios to find the triangle's hypotenuse and its sine, cosine, and tangent of \(45^{\circ}\).

• The hypotenuse is the longest side opposite the right angle.
• Each of the smaller angles in an isosceles right triangle is \(45^{\circ}\).

This symmetry not only makes calculations easier but also reinforces fundamental trigonometric principles.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that applies to right triangles. It states that the square of the hypotenuse (c) of a right triangle is equal to the sum of the squares of the other two sides (a and b), expressed as \(a^2 + b^2 = c^2\).

For an isosceles right triangle where both legs are of equal length, say 3 units each, we apply this theorem to find the hypotenuse. By plugging the values of the legs into the equation, we get:

• \(3^2 + 3^2 = c^2\)
• This simplifies to \(9 + 9 = c^2\)
• Resulting in \(c^2 = 18\) or \(c = \sqrt{18} = 3\sqrt{2}\)

These calculations are key in determining the exact trigonometric ratios of \(45^{\circ}\). Understanding this theorem is crucial for solving problems involving right triangles, especially in standardized tests and assignments.

Rationalizing the Denominator

Rationalizing the denominator is a process used in mathematics to eliminate radicals, such as square roots, from the bottom of a fraction. This operation helps in simplifying the expression and is more aesthetically pleasing and easier to interpret.

In the context of finding trigonometric ratios using an isosceles right triangle, after determining the hypotenuse as \(3\sqrt{2}\), we find the trigonometric function values, such as sine and cosine, by evaluating the ratios using this hypotenuse.

• For \(\sin 45^{\circ}\), the calculation is \(\frac{3}{3\sqrt{2}}\).
• To rationalize, multiply both the numerator and the denominator by \(\sqrt{2}\) to get \(\frac{3\sqrt{2}}{6}= \frac{\sqrt{2}}{2}\).

This gives us a simplified exact value, which is often required in math problems. Rationalizing is a widely-used technique, ensuring expressions are in their simplest and most recognizable form.