Trigonometric Equations and Identities Quiz

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10 Questions

How can the equation $\sin (t)=0$ be solved using unit circle values?

The equation $\sin (t)=0$ can be solved using unit circle values by identifying the angles where $\sin (t)=0$, which are $t=0, \pi , 2\pi$, and so on.

Explain the technique used to solve the equation $2x^{2} +x=0$.

The technique used to solve the equation $2x^{2} +x=0$ is factoring. We factor the equation to get $x(2x+1)=0$, and then find the solutions $x = 0$ or $x = -\dfrac{1},{2}$.

What algebraic techniques are used to solve polynomial trig functions?

Algebraic techniques such as factoring and the quadratic formula, along with trigonometric identities and techniques, are used to solve polynomial trig functions.

What is the equation created by the composition of the functions $f(x)=2x^{2} +x$ and $g(t)=\sin (t)$?

The equation created by the composition of the functions $f(x)=2x^{2} +x$ and $g(t)=\sin (t)$ is $f(g(t))=2(\sin (t))^{2} +(\sin (t))=2\sin ^{2} (t)+\sin (t)$.

Solve the equation $3\sec ^{2} (t)-5\sec (t)-2=0$ for all solutions with $0\le t$.

The solutions for the equation $3\sec ^{2} (t)-5\sec (t)-2=0$ with $0\le t$ are $t = \dfrac{\pi},{3}$ and $t = \dfrac{5\pi},{3}$.

In trigonometry, what are trigonometric identities and what is their significance in simplifying expressions involving trigonometric functions?

Trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. They are significant in simplifying expressions involving trigonometric functions, making it easier to work with such expressions.

What is the Pythagorean identity in trigonometry and how is it related to the Pythagorean theorem?

The Pythagorean identity in trigonometry is the basic relationship between the sine and cosine, represented as $\sin^2\theta + \cos^2\theta = 1$. This identity is related to the Pythagorean theorem as it can be viewed as a version of the theorem and follows from the equation $x^2 + y^2 = 1$ for the unit circle.

How can the Pythagorean identity be used to solve for either the sine or the cosine, and what is the significance of the sign in this context?

The Pythagorean identity can be solved for either the sine or the cosine by rearranging the equation to express one of the functions in terms of the other. The significance of the sign depends on the quadrant of the angle $\theta$.

What is the geometric interpretation of the Pythagorean identity in trigonometry?

The geometric interpretation of the Pythagorean identity is that it relates the squares of the sine and cosine of an angle to a constant value. This can be visualized as a fundamental relationship between the lengths of sides in a right-angled triangle and the radius of the unit circle.

What is the practical application of trigonometric identities, particularly the Pythagorean identity, in the context of integrating non-trigonometric functions?

A practical application of trigonometric identities, such as the Pythagorean identity, is in the integration of non-trigonometric functions. A common technique involves using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity to make the integration process more manageable.

Study Notes

Trigonometric Equations

  • The equation $\sin (t) = 0$ can be solved using unit circle values by finding the angles on the unit circle where the sine is zero, which are 0, π, and multiples of 2π.

Quadratic Equations

  • The equation $2x^2 + x = 0$ can be solved using factorization or the quadratic formula.

Polynomial Trigonometric Functions

  • Algebraic techniques such as factoring, synthetic division, and the rational roots theorem are used to solve polynomial trig functions.

Composition of Functions

  • The composition of functions $f(x) = 2x^2 + x$ and $g(t) = \sin (t)$ results in a new function $f(g(t)) = 2\sin^2 (t) + \sin (t)$.

Trigonometric Identities

  • Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable(s), allowing for the simplification of expressions involving trigonometric functions.
  • The Pythagorean identity is a fundamental trigonometric identity stating that $\sin^2 (t) + \cos^2 (t) = 1$.

Pythagorean Identity

  • The Pythagorean identity is related to the Pythagorean theorem, as both are based on the concept of right triangles.
  • The Pythagorean identity can be used to solve for either the sine or the cosine by rearranging the equation to isolate the desired function.
  • The sign of the sine and cosine functions is significant, as it affects the quadrant of the angle in the unit circle.

Geometric Interpretation

  • The geometric interpretation of the Pythagorean identity is that the sum of the squares of the sine and cosine of an angle is equal to 1, which represents the fact that a right triangle with legs of length $\sin (t)$ and $\cos (t)$ has a hypotenuse of length 1.

Practical Applications

  • Trigonometric identities, particularly the Pythagorean identity, have practical applications in integrating non-trigonometric functions, as they allow for the simplification of complex expressions.

Test your understanding of solving trigonometric equations with identities with this quiz. Practice applying trigonometric identities to solve various equations and sharpen your skills in trigonometry.

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