# Determine Whether Trigonometry Functions $\sin^2(x), \cos^2(x), 1$ are Linearly Independent or Dependent

## Problem 365

Let $f(x)=\sin^2(x)$, $g(x)=\cos^2(x)$, and $h(x)=1$. These are vectors in $C[-1, 1]$.

Determine whether the set $\{f(x), \, g(x), \, h(x)\}$ is linearly dependent or linearly independent.

(*The Ohio State University, Linear Algebra Midterm Exam Problem*)

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## Solution.

We claim that the set is linearly dependent. To show the claim, we need to find nontrivial scalars $c_1, c_2, c_3$ such that

\[c_1 f(x)+c_2 g(x)+ c_3 h(x)=0.\]

From trigonometry, we know the identity

\[\sin^2(x)+\cos^2(x)=1.\]
This implies that we have

\[\sin^2(x)+\cos^2(x)-1=0.\]

So we can choose $c_1=1, c_2=1, c_3=-1$, and thus the set is linearly dependent.

## Linear Algebra Midterm Exam 2 Problems and Solutions

- True of False Problems and Solutions: True or False problems of vector spaces and linear transformations
- Problem 1 and its solution: See (7) in the post “10 examples of subsets that are not subspaces of vector spaces”
- Problem 2 and its solution (current problem): Determine whether trigonometry functions $\sin^2(x), \cos^2(x), 1$ are linearly independent or dependent
- Problem 3 and its solution: Orthonormal basis of null space and row space
- Problem 4 and its solution: Basis of span in vector space of polynomials of degree 2 or less
- Problem 5 and its solution: Determine value of linear transformation from $R^3$ to $R^2$
- Problem 6 and its solution: Rank and nullity of linear transformation from $R^3$ to $R^2$
- Problem 7 and its solution: Find matrix representation of linear transformation from $R^2$ to $R^2$
- Problem 8 and its solution: Hyperplane through origin is subspace of 4-dimensional vector space

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[…] Problem 2 and its solution: Determine whether trigonometry functions $sin^2(x), cos^2(x), 1$ are linearly independent or dependent […]

[…] Problem 2 and its solution: Determine whether trigonometry functions $sin^2(x), cos^2(x), 1$ are linearly independent or dependent […]