Problem 60 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 57-62: Use an addition or subtraction formula to find the solutions of the equation that are in the interval \([0, \pi\) ). $$ \sin 3 t \cos t+\cos 3 t \sin t=-\frac{1}{2} $$

Step-by-Step Solution

Answer

The solutions are \( t = \frac{7\pi}{24} \) and \( t = \frac{11\pi}{24} \).

Step 1: Recognize the Formula

The equation can be expressed in the form of the sine addition formula. Recall that the sum of sine angles formula is: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \).

Step 2: Apply the Sine Addition Formula

Identify \( a = 3t \) and \( b = t \). The left side of the equation becomes: \( \sin(3t + t) = \sin 4t \). The equation is now \( \sin 4t = -\frac{1}{2} \).

Step 3: Solve the Equation

We need to find values of \( t \) such that \( \sin 4t = -\frac{1}{2} \). Since \( \sin \theta = -\frac{1}{2} \) when \( \theta = \frac{7\pi}{6} \) or \( \frac{11\pi}{6} \), set \( 4t = \frac{7\pi}{6} + 2k\pi \) or \( 4t = \frac{11\pi}{6} + 2k\pi \) for integers \( k \).

Step 4: Evaluate in Given Interval

Solve for \( t \) by dividing both sides by 4. For \( 4t = \frac{7\pi}{6} + 2k\pi \), \( t = \frac{7\pi}{24} + \frac{k\pi}{2} \) and for \( 4t = \frac{11\pi}{6} + 2k\pi \), \( t = \frac{11\pi}{24} + \frac{k\pi}{2} \).

Step 5: Determine Valid Solutions

Check which solutions are within the interval \([0, \pi)\):- Starting with \( k = 0 \), \( t = \frac{7\pi}{24} \) and \( t = \frac{11\pi}{24} \) are both within the interval. - With \( k = 1 \), the values will exceed \( \pi \), confirming no other values are within the interval.

Key Concepts

sine addition formula

The sine addition formula is a critical tool in solving trigonometric equations involving sums of angles. This formula helps us express the sine of a sum of two angles, say \( a \) and \( b \), as a combination of the sines and cosines of the individual angles. The formula is given by: \[ \sin(a + b) = \sin a \cos b + \cos a \sin b \] This formula becomes exceptionally useful when working with equations where combined angles appear.

• It allows the simplification of expressions.
• It provides a way to transform complex expressions into more manageable forms.
• It aids in recognizing patterns within trigonometric problems.

In the exercise, the given expression matches the structure of the sine addition formula, which transforms the original equation into a simpler form \( \sin 4t = -\frac{1}{2} \). This new form allows us to find solutions using known angles on the unit circle.

trigonometric identities

Trigonometric identities are equations that hold true for all possible values of the involved angles. These identities are foundational tools in trigonometry, providing ways to manipulate and simplify trigonometric expressions.

• Pythagorean identities: Like \( \sin^2 x + \cos^2 x = 1 \).
• Angle sum and difference identities: Such as \( \sin(a \pm b) \), which includes additions like our exercise.
• Double angle identities: For instance, \( \sin 2a = 2 \sin a \cos a \).

Using these identities can simplify solving equations and verifying expressions. In our exercise, recognizing that the expression fits into the sine addition formula enabled us to simplify the equation substantially, leading to known results of specific angle resolutions on the unit circle.

unit circle

The unit circle is essential for understanding trigonometric functions and their relationships with angles. It is a circle with a radius of one, centered at the origin of a coordinate plane. The unit circle helps visualize the values of trigonometric functions for any given angle.

• Sine corresponds to the \( y \)-coordinate of a point on the circle.
• Cosine aligns with the \( x \)-coordinate of that point.
• Common angles: \( 0 \), \( \pi/6 \), \( \pi/4 \), \( \pi/3 \), \( \pi/2 \), etc.

In our solved problem, the unit circle provides specific angles where \( \sin \theta \) achieves a value of \(-\frac{1}{2} \), namely \( \theta = \frac{7\pi}{6} \) and \( \theta = \frac{11\pi}{6} \). Recognizing these angles is crucial, as they help determine solutions within a specific interval.

interval notation

Interval notation is a mathematical method used to describe ranges of values, often seen with solutions to equations or inequalities. This notation provides a clear, efficient way of expressing the domain of solutions.

• Brackets \( [] \) denote inclusivity, meaning the endpoint is included.
• Parentheses \( () \) indicate exclusivity, excluding the endpoint.
• For instance, \([0, \pi)\) implies the range from 0 up to but not including \( \pi \).

In the example, the interval \([0, \pi)\) defines the range within which we seek solutions for \( t \). This ensures solutions like \( t = \frac{7\pi}{24} \) and \( t = \frac{11\pi}{24} \) remain valid, as they fall within the specified interval. Understanding interval notation aids in expressing your results clearly for exams and communicating mathematical ranges succinctly.