Which of the following are identities? Check all that apply.
In mathematics, identities are fundamental concepts that represent equations that are always true. They are essential tools for simplifying expressions, solving equations, and understanding the properties of mathematical operations. In this article, we will explore some common identities and determine which of them are indeed identities.
1. Pythagorean Theorem
The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written as:
\[ a^2 + b^2 = c^2 \]
where \( a \) and \( b \) are the lengths of the legs, and \( c \) is the length of the hypotenuse. This identity is always true and applies to all right-angled triangles. Therefore, it is an identity.
2. Complementary and Supplementary Angles
Complementary angles are two angles that add up to 90 degrees, while supplementary angles add up to 180 degrees. The following identities describe these relationships:
\[ \text{Complementary Angles: } \alpha + \beta = 90^\circ \]
\[ \text{Supplementary Angles: } \alpha + \beta = 180^\circ \]
These identities are always true, as the sum of two complementary angles will always be 90 degrees, and the sum of two supplementary angles will always be 180 degrees. Hence, they are identities.
3. Binomial Theorem
The Binomial Theorem is a formula that expands the power of a binomial (a sum or difference of two terms) raised to a non-negative integer power. The formula is:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
where \( \binom{n}{k} \) is the binomial coefficient. This identity is always true for any real numbers \( a \) and \( b \) and any non-negative integer \( n \). Therefore, it is an identity.
4. Quadratic Formula
The Quadratic Formula is a formula used to find the roots of a quadratic equation of the form \( ax^2 + bx + c = 0 \). The formula is:
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
This formula is always true for any quadratic equation, as it provides the exact solutions for the roots of the equation. Hence, it is an identity.
5. Logarithmic Identity
The logarithmic identity states that:
\[ \log_a (a^x) = x \]
where \( a \) is the base of the logarithm, and \( x \) is any real number. This identity is always true, as the logarithm of \( a \) raised to the power of \( x \) is indeed \( x \). Therefore, it is an identity.
In conclusion, all the identities mentioned above are indeed identities. They are fundamental concepts in mathematics that are always true and have wide applications in various fields. Always keep these identities in mind while solving mathematical problems, as they can simplify your work and provide insights into the properties of mathematical operations.
