Question

Show \(\cos \left(180^{\circ}+\theta\right)=-\cos \theta\)

Short answer

Question: Prove the trigonometric identity \(\cos(180^{\circ}+\theta)=-\cos \theta\). Answer: By applying the sum of angles formula and simplifying the result, we proved that \(\cos(180^{\circ}+\theta)=-\cos \theta\).

Step-by-step solution

Use the Sum of Angles formula

Let's take the left-hand side of the equation and use the formula for the cosine of the sum of two angles. According to the formula, \(\cos(A + B) = \cos A \cos B - \sin A \sin B.\) Replacing A with \(180^{\circ}\) and B with \(\theta\), we get: \(\cos \left(180^{\circ}+\theta\right)=\cos 180^{\circ} \cos \theta - \sin 180^{\circ} \sin \theta\)

Determine the value of cosine and sine

The cosine of \(180^{\circ}\) is \(-1\) and the sine of \(180^{\circ}\) is \(0.\) Substitute these values in the equation: \(\cos(180^{\circ}+\theta)= -1 \cdot \cos \theta - 0 \cdot \sin \theta\)

Simplify the result

Simplifying the expression obtained, we get: \(\cos(180^{\circ}+\theta)=-\cos \theta - 0\), which simplifies to \(\cos(180^{\circ}+\theta)=-\cos \theta\)

Compare with the right-hand side

Now, observe that left-hand side is exactly equal to the right-hand side. This implies that the given trigonometric identity is correct. Therefore, we have successfully proved the identity \(\cos(180^{\circ}+\theta)=-\cos \theta\) using the sum of angles formula in trigonometry.

Key concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions, often abbreviated as "cos." It is crucial for understanding the relationships between angles and lengths in right-angled triangles. For a given angle \(\theta\), the cosine value represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
This function is particularly important because its properties allow us to simplify complex trigonometric expressions. For instance, cosine has a periodicity of \(360^{\circ}\) or \(2\pi\) radians, meaning \(\cos(\theta + 360^{\circ}) = \cos \theta\). Understanding these properties is essential for solving various problems in trigonometry, such as the one presented in the exercise.

Sum of Angles

Trigonometry often requires us to calculate the cosine or sine of an angle that is expressed as the sum of two other angles. This is where the sum of angles formula becomes applicable. The cosine sum formula is expressed as \(\cos(A + B) = \cos A \cos B - \sin A \sin B\).
By using this formula, we can transform the operation of finding the cosine of a sum into a product operation between the trigonometric functions of individual angles. This transformation significantly simplifies calculations and helps prove identities, like \(\cos(180^{\circ} + \theta) = -\cos \theta\), as demonstrated in the exercise.

• The process generally involves:
  • Identifying the angles involved
  • Applying the sum of angles formula
  • Simplifying with known values of specific angles, such as \(\cos 180^{\circ} = -1\) and \(\sin 180^{\circ} = 0\)

Using this formula is a powerful technique in trigonometry for proving and simplifying expressions.

Trigonometry

Trigonometry is the study of relationships between the sides and angles of triangles. It is a branch of mathematics that has applications in various fields, including physics, engineering, and architecture.
The core functions of trigonometry—sine, cosine, and tangent—allow us to express and manipulate angular measures and are fundamental to analytical geometry.
In this context, we focus on using trigonometric identities, such as the sum of angles formula, which plays a central role in simplifying expressions and solving equations like the one presented in the exercise. These identities help us explore deeper mathematical concepts and solve real-world problems efficiently.

• Key Trigonometric Identities include:
  • Pythagorean Identity: \(\sin^2 \theta + \cos^2 \theta = 1\)
  • Angle Sum and Difference Identities
  • Double Angle Formulas

Mastering these identities can enhance one's ability to analyze and solve a wide range of mathematical problems.

Proofs in Mathematics

In mathematics, proofs are logical arguments that demonstrate the truth of a given statement. They are essential for establishing the validity of mathematical concepts and identities, like those in trigonometry.
Mathematical proofs often involve a series of logical steps that lead from known truths to the conclusion we want to establish.

• Steps often include:
  • Assuming known values or identities
  • Applying mathematical operations or transformations
  • Simplifying the expressions to arrive at the identity

In the exercise at hand, we proved \(\cos(180^{\circ} + \theta) = -\cos \theta\) by utilizing established identities and simplifying the equation. This process reinforces the reliability of mathematical reasoning and helps students understand how to apply these techniques to various problems. Proofs not only confirm that identities hold true but also deepen our comprehension of the relationships within mathematics, enhancing problem-solving skills.