Question

Show \(\sin \left(\frac{\pi}{2}-\theta\right)=\cos \theta\).

Short answer

Question: Prove the following trigonometric identity: \(\sin \left(\frac{\pi}{2}-\theta\right)=\cos \theta\) Answer: Using the co-function identity formula \(\sin \left(\frac{\pi}{2} - x\right) = \cos(x)\), we can plug in our expression and confirm the result as follows: \(\sin \left(\frac{\pi}{2}-\theta\right) = \cos(\theta)\). This verifies that \(\sin \left(\frac{\pi}{2}-\theta\right)=\cos \theta\).

Step-by-step solution

Recall the co-function identity formula

In this case, we will be using the following co-function identity formula: \(\sin \left(\frac{\pi}{2} - x\right) = \cos(x)\) The formula demonstrates that the sine of the complementary angle to x (90 degrees, or pi/2 radians, minus x) is equal to the cosine of x.

Apply the co-function identity to our expression

Now that we know the co-function identity formula, let's apply it to the expression we have to prove. We are trying to show that \(\sin \left(\frac{\pi}{2}-\theta\right)=\cos \theta\). We can just plug in the formula for the sine of the complementary angle: \(\sin \left(\frac{\pi}{2}-\theta\right) = \cos(\theta)\)

Confirm the result

Based on the co-function identity formula, our expression holds true: \(\sin \left(\frac{\pi}{2}-\theta\right) = \cos \theta\) That demonstrates that \(\sin \left(\frac{\pi}{2}-\theta\right)=\cos \theta\). The co-function identity allowed us to connect the sine and cosine functions and prove the given identity.

Key concepts

Co-Function Identity

In trigonometry, the co-function identity is a fundamental concept. It establishes relationships between sine and cosine functions for complementary angles. Complementary angles are two angles whose sum is 90 degrees, or \(\frac{\pi}{2}\) radians.

A co-function identity specifically tells us that the sine of one angle is equal to the cosine of its complementary angle. In mathematical terms, it is expressed as follows: \[\sin \left(\frac{\pi}{2} - x\right) = \cos(x)\].

In simpler terms, if you know the sine of an angle, you can easily find the cosine of its complementary angle using this identity. It provides a direct way to interchange sine and cosine values through simple angle transformations. This identity was the basis for solving our initial problem.

Sine and Cosine Functions

Sine and cosine are two of the primary trigonometric functions utilized in mathematics to understand the properties of triangles and oscillatory motions. These functions are mainly associated with angles in a right-angled triangle.

**The Sine Function:**

• For a given angle \(\theta\), sine (\(\sin(\theta)\)) is calculated as the ratio of the length of the opposite side to the hypotenuse in a right triangle.
• It represents vertical displacement in circular motion when considering the unit circle.

**The Cosine Function:**

• Cosine (\(\cos(\theta)\)) of an angle is the ratio of the length of the adjacent side to the hypotenuse in a right triangle.
• In the unit circle, it demonstrates horizontal displacement.

Together, sine and cosine functions describe periodic phenomena such as sound waves and light. Understanding their complementary nature, as seen with co-function identities, reveals much about their collective behavior.

Complementary Angles

Angles are considered complementary if their sum is precisely 90 degrees or \(\frac{\pi}{2}\) radians. In this way, complementary angles are tightly interlinked, particularly in the context of trigonometric functions.

**Why Complementary Angles Matter:**

• They help establish relationships between different trigonometric functions using formulas like co-function identities.
• Allow easy transformation between trigonometric values, helping in solving various geometrical problems.

For instance, knowing that two angles \(\theta\) and \(\frac{\pi}{2} - \theta\) are complementary allows us to apply the identity \(\sin \left(\frac{\pi}{2} - \theta\right) = \cos(\theta)\).
These types of relationships highlight why complementary angles are an integral part of trigonometry, simplifying complex problems and providing greater insight into function behavior.