Understanding the Trigonometric Value of sin45°

Understanding the value of sin45°sin45^circ as 121/sqrt{2} is crucial in trigonometry. This article explains this formula using geometric properties and the Pythagorean theorem.

Explanation of Trigonometric Properties of a 45-45-90 Triangle

A 45-45-90 triangle, also known as an isosceles right triangle, has several unique properties. One of the angles is 45°, and another is 90°, making the triangle isosceles.

Triangle Properties

In a 45-45-90 triangle:

The two legs are equal in length. The hypotenuse can be calculated using the Pythagorean theorem.

Pythagorean Theorem Application

Let the lengths of the legs be 1. Using the Pythagorean theorem:

begin{align*} text{Hypotenuse} sqrt{1^2 1^2} sqrt{2} end{align*}

Sine Definition

The sine of an angle in a right triangle is the ratio of the length of the opposite side to the length of the hypotenuse.

Sine of 45°

For a 45-45-90 triangle:

The opposite side to 45° is 1 (one leg of the triangle). The hypotenuse is 2sqrt{2}.

Therefore:

begin{align*} sin45^circ frac{text{opposite}}{text{hypotenuse}} frac{1}{sqrt{2}} end{align*}

Rationalizing the Denominator

It's common to express 121/sqrt{2} in a rationalized form:

begin{align*} frac{1}{sqrt{2}} cdot frac{sqrt{2}}{sqrt{2}} frac{sqrt{2}}{2} end{align*}

Hence, we have:

sin45^circ frac{1}{sqrt{2}}sin45^circ frac{1}{sqrt{2}} and sin45^circ frac{sqrt{2}}{2}sin45^circ frac{sqrt{2}}{2}.

Both forms are widely used in trigonometry.

The Sine Function and Its Behavior

The sin function is similar to any other curve. Its y-axis values oscillate between 0 and 1. For simplification, let's consider the x-axis range [0, 90°].

begin{align*} sin0^circ0 sin90^circ1 end{align*}

Since the sin function is not linear, its value at 45° is not 121/2. If it were, the function would be a straight line with a 45° slope, which is not the case.

Therefore, the value of sin45^circsin45^circ is defined by the properties of a perfect isosceles right triangle.

Verification Using Triangle Properties

Let's consider another approach to prove the value of sin45^circsin45^circ.

Isosceles Right-Angled Triangle

In an isosceles right-angled triangle, two of the angles are 45°.

Let the two equal sides be xx. Using the Pythagorean theorem:

begin{align*} h^2 x^2 b^2 x^2 x^2 2x^2 h sqrt{2}x frac{x}{h} frac{x}{sqrt{2}x} frac{1}{sqrt{2}} end{align*}

Hence, we have:

sin45^circ frac{1}{sqrt{2}}sin45^circ frac{1}{sqrt{2}}.