Question

Angle \(x\) is one of the acute angles in a right triangle. If the measure of angle \(x\) is \(30^{\circ},\) what is the value of \((\sin x)^{2}+(\cos x)^{2} ?\) a.\(\frac{1}{4}\) b.\(\frac{1}{2}\) c.1 d.2

Short answer

1

Step-by-step solution

Find \(\sin x\) and \(\cos x\)

For angle \(\x\) equal to \(\30^{\circ}\), \(\sin 30^{\circ} = \frac{1}{2}\) and \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\).

Calculate \(\sin x\)^{2}\)

Square the value of \(\sin x\): \(\left( \sin 30^{\circ} \right)^{2} = \left( \frac{1}{2} \right)^{2} = \frac{1}{4}\).

Calculate \(\cos x\)^{2}\)

Square the value of \(\cos x\): \(\left( \cos 30^{\circ} \right)^{2} = \left( \frac{\sqrt{3}}{2} \right)^{2} = \frac{3}{4}\).

Sum the squares of \(\sin x\) and \(\cos x\)

Add the squares obtained in steps 2 and 3: \(\left( \frac{1}{4} \right) + \left( \frac{3}{4} \right) = 1\).

Key concepts

headline of the respective core concept

To understand the trigonometric properties in a right triangle, we first need to go over what a right triangle really is. A right triangle is a triangle in which one of the angles is exactly 90 degrees.
This 90-degree angle is called the right angle. The side opposite the right angle is known as the hypotenuse, which is the longest side of the triangle. The other two angles in the triangle are acute angles – they each measure less than 90 degrees.
The sides across from these acute angles are referred to as the 'legs' of the triangle. In trigonometry, we often focus on the relationships between these sides and the angles.

headline of the respective core concept

Acute angles are angles that measure less than 90 degrees. In the context of a right triangle, these are the two angles other than the right angle.
It's important to remember that in any triangle, the sum of all angles is always 180 degrees. Therefore, the two acute angles in a right triangle always add up to 90 degrees.
For example, if one angle in a right triangle is 30 degrees (like angle x in the problem), the other acute angle must be 60 degrees (since 90 - 30 = 60). Understanding this relationship helps when working with trigonometric identities, which we'll discuss next.

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Trigonometric identities are equations that are true for all angles in a right triangle. These identities are foundational in trigonometry, helping to solve various problems.
A key identity involving sine and cosine is: \(\text{sine}^{2}(x) + \text{cosine}^{2}(x) = 1\). This tells us that if we square the sine and cosine of any angle and add them together, the result will always be 1.
For angle x in our exercise, if \(\text{sine}(30^\text{degrees})\) equals \(\frac{1}{2}\) and \(\text{cosine}(30^\text{degrees}) = \frac{\text{\textbackslash sqrt{3}}}{2}\), we can check this identity by squaring these values and adding them.
This principle makes calculations straightforward and verifies the integrity of trigonometric functions.

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To solve trigonometric problems correctly, understanding the functions of sine and cosine is crucial.
The sine of an angle in a right triangle is the ratio of the length of the side opposite to the angle over the hypotenuse.
For example, \(\text{sine}(30^\text{degrees}) = \frac{1}{2}\) because in a 30-60-90 triangle, the opposite side to the 30 degrees angle is half the hypotenuse.
The cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse. \(\text{cosine}(30^\text{degrees}) = \frac{\text{\textbackslash sqrt{3}}}{2}\) because the adjacent side in this scenario is \(\text{\textbackslash sqrt{3}}/2\) of the hypotenuse. Using these functions alongside the fundamental identity of \(\text{sine}^{2} + \text{cosine}^{2}\) helps solve trigonometric equations.