What is the Relation between Sine and Cosine Graphs?

S6-SA2-0263

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The relation between sine and cosine graphs is that they are essentially the same 'wave' shape, but shifted horizontally. The cosine graph is simply the sine graph moved to the left by 90 degrees (or pi/2 radians).

Simple Example

Quick Example

Imagine you have two identical cricket matches happening, but one match starts 30 minutes later than the other. If you plot the 'excitement level' over time for both matches, the graphs would look very similar, just one would be shifted later. Similarly, sine and cosine graphs are just shifted versions of each other.

Worked Example

Step-by-Step

Let's see how sin(x) and cos(x) values relate at specific angles.

STEP 1: Recall the basic values: sin(0) = 0, cos(0) = 1.
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STEP 2: Recall sin(90 degrees) = 1, cos(90 degrees) = 0.
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STEP 3: Now, let's compare sin(x) with cos(x - 90 degrees). If x = 90 degrees, then cos(90 - 90) = cos(0) = 1. This is equal to sin(90).
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STEP 4: If x = 0 degrees, then cos(0 - 90) = cos(-90). We know cos(-angle) = cos(angle), so cos(-90) = cos(90) = 0. This is equal to sin(0).
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STEP 5: This shows that the value of sin(x) is the same as cos(x - 90 degrees). This means the sine graph is the cosine graph shifted 90 degrees to the right.
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STEP 6: Alternatively, the cosine graph is the sine graph shifted 90 degrees to the left. So, cos(x) = sin(x + 90 degrees).

ANSWER: The sine and cosine graphs are horizontally shifted versions of each other, specifically, cos(x) = sin(x + 90 degrees) or sin(x) = cos(x - 90 degrees).

Why It Matters

Understanding this shift is crucial in fields like Physics to describe waves (light, sound) and oscillations. Engineers use this to design signals in mobile phones and radio. It's also fundamental in computer graphics for creating smooth movements and animations.

Common Mistakes

MISTAKE: Thinking sine and cosine are completely different shapes. | CORRECTION: Remember they are the exact same wave shape, just one starts its cycle at a different point than the other.

MISTAKE: Confusing the direction of the shift (left vs. right). | CORRECTION: Visualize or remember: cos(x) 'starts' at its peak (at x=0), like sin(x) would after being shifted LEFT by 90 degrees to reach its peak earlier.

MISTAKE: Using degrees for some parts of a problem and radians for others without converting. | CORRECTION: Always ensure you are consistent with units. If the shift is given as pi/2, then your angles should also be in radians, or convert pi/2 to 90 degrees.

Practice Questions

Try It Yourself

QUESTION: If the sine graph crosses the x-axis at 0, 180, and 360 degrees, where will the cosine graph cross the x-axis in the range 0 to 360 degrees? | ANSWER: 90 and 270 degrees

QUESTION: A sound wave can be described by a sine function. If we want to represent the same sound wave but starting its cycle 90 degrees earlier, which trigonometric function would be more appropriate? | ANSWER: Cosine function (since cos(x) is sin(x) shifted 90 degrees left, meaning it 'starts' earlier)

QUESTION: The maximum value of sin(x) is 1. At what angle 'x' (between 0 and 360 degrees) does cos(x) reach its maximum value, and how does this relate to the sine graph? | ANSWER: cos(x) reaches its maximum value of 1 at x = 0 degrees (and 360 degrees). This is where the sine graph is 0, showing a 90-degree phase difference.

MCQ

Quick Quiz

Which of the following statements correctly describes the relationship between the sine and cosine graphs?

The cosine graph is the sine graph shifted 90 degrees to the right.

The sine graph is the cosine graph shifted 90 degrees to the right.

The sine and cosine graphs are completely different shapes.

The cosine graph has double the amplitude of the sine graph.

The Correct Answer Is:

Option B is correct because sin(x) = cos(x - 90 degrees), meaning the sine graph is the cosine graph shifted 90 degrees to the right. Option A describes the opposite shift. Options C and D are incorrect as they are the same shape and have the same amplitude.

Real World Connection

In the Real World

In India, electrical engineers working on power grids use sine and cosine waves to model alternating current (AC) electricity. Understanding their phase difference (the 90-degree shift) is vital for efficient power transmission and managing voltage fluctuations across cities and villages.

Key Vocabulary

Key Terms

PHASE SHIFT: A horizontal displacement of a periodic function's graph. | PERIODIC FUNCTION: A function that repeats its values in regular intervals or periods. | AMPLITUDE: The maximum displacement or distance moved by a point on a vibrating body or wave measured from its equilibrium position. | RADIANS: A unit of angle, equal to the angle subtended at the center of a circle by an arc equal in length to the radius.

What's Next

What to Learn Next

Now that you understand how sine and cosine graphs relate, you're ready to explore 'Trigonometric Identities'. These are equations involving trigonometric functions that are true for every value of the variable, and they build directly on this understanding of shifts and relationships.