Can You Solve This Trigonometric Equation for $x$?

In summary, a trigonometric equation is an equation that involves trigonometric functions and an unknown variable, with the goal of finding the value(s) of the variable that make the equation true. The basic trigonometric identities include the Pythagorean identities and other important identities such as the reciprocal, quotient, and even-odd identities. To solve a trigonometric equation, you can use algebraic methods and trigonometric identities. The domain of a trigonometric equation refers to the set of all possible values for the independent variable, and trigonometric equations have various real-world applications in fields such as engineering, physics, and architecture.
  • #1
anemone
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Solve for $x$ such that $2\sin(x+30^\circ)\sin 16^\circ \sin 76^\circ=\sin 2028^\circ \sin 210^\circ$ for $0\lt x \lt 180^\circ$.
 
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  • #2
anemone said:
Solve for $x$ such that $2\sin(x+30^\circ)\sin 16^\circ \sin 76^\circ=\sin 2028^\circ \sin 210^\circ$ for $0\lt x \lt 180^\circ$.

\(\displaystyle \sin2028^\circ\sin210^\circ=-\dfrac{\sin228^\circ}{2}=\dfrac{\sin48^\circ}{2}\)

\(\displaystyle \sin48^\circ=4\sin(x+30^\circ)\sin16^\circ\sin76^\circ\)

\(\displaystyle 3\sin16^\circ-4\sin^316^\circ=4\sin(x+30^\circ)\sin16^\circ\sin76^\circ\)

\(\displaystyle 1+2\cos32^\circ=4\sin(x+30^\circ)\sin76^\circ\)

\(\displaystyle 1+2\cos32^\circ=2(\cos(x-46^\circ)-\cos(x+106^\circ))\)

\(\displaystyle \text{By inspection, }x=14^\circ\)
 
  • #3
anemone said:
Solve for $x$ such that $2\sin(x+30^\circ)\sin 16^\circ \sin 76^\circ=\sin 2028^\circ \sin 210^\circ$ for $0\lt x \lt 180^\circ$.

$2\sin(x+30^\circ)\sin 16^\circ \sin 76^\circ=\sin 2028^\circ \sin 210^\circ$
= $\sin (360^\circ * 5 + 180^\circ + 48^\circ) \sin (180^\circ +30^\circ)$
= $(-\sin\, 48^\circ) (-\sin \,30^\circ)$
= $\dfrac{\sin\,48^\circ}{2}$
= $\dfrac{\sin\, 3*16^\circ}{2}$
= $\dfrac{3\sin\,16^\circ-4\sin ^3 16^\circ}{2}$
hence
$2\sin(x+30^\circ)\sin 16^\circ \sin 76^\circ = \sin\,16^\circ \dfrac{3-4\sin ^2 16^\circ}{2}$
or
$4\sin(x+30^\circ)\sin 76^\circ = 3-4\sin ^2 16^\circ$
= $ 1 + 2(1-2\sin^2 16^\circ)$
= $ 1+ 2 \cos\,32^\circ$
= $(2(\cos\,60^\circ +2 \cos\,32^\circ)$
= $ 2 (2 \cos\,46^\circ \cos\,14^\circ)$
= $ 4 \cos\,46^\circ \sin \,76^\circ$
hence
$\sin(x+30^\circ)=\cos\,46^\circ=\sin\,44^\circ$
or $x=14^\circ$

edit there is a solution I missed based on comment below
in the 2nd quadrant
$\sin(x+30^\circ)=\sin\,136^\circ$
so $x= 106^\circ$
 
Last edited:
  • #4
Thanks both for participating and the solution...but...

Are you certain you haven't missed any solution? (Mmm)
 
  • #5
anemone said:
Thanks both for participating and the solution...but...

Are you certain you haven't missed any solution? (Mmm)
I missed the solution $x + 30^\circ = 136^\circ$ or $x = 106^\circ$
 

FAQ: Can You Solve This Trigonometric Equation for $x$?

What is a trigonometric equation?

A trigonometric equation is an equation that involves one or more trigonometric functions, such as sine, cosine, or tangent, and an unknown variable. The goal of solving a trigonometric equation is to find the value(s) of the variable that make the equation true.

What are the basic trigonometric identities?

The basic trigonometric identities include the Pythagorean identities, which relate the three main trigonometric functions (sine, cosine, and tangent) to each other and to the unit circle. Other important identities include the reciprocal identities, quotient identities, and even-odd identities.

How do you solve a trigonometric equation?

To solve a trigonometric equation, you can use algebraic methods, such as factoring, combining like terms, and isolating the variable. You can also use trigonometric identities to simplify the equation and make it easier to solve. In some cases, you may need to use a calculator or graphing software to find the solutions.

What is the domain of a trigonometric equation?

The domain of a trigonometric equation refers to the set of all possible values for the independent variable (usually denoted as x) that make the equation defined. In general, the domain of a trigonometric equation is all real numbers, but there may be restrictions depending on the specific trigonometric function and equation.

What are some real-world applications of trigonometric equations?

Trigonometric equations have many real-world applications, including in engineering, physics, architecture, and navigation. For example, trigonometric equations can be used to calculate the height of a building, the distance between two objects, or the angles of a triangle. They are also essential in solving problems involving waves, vibrations, and periodic phenomena.

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