Example: Intersection of sine and cosine | Graphs of trig functions | Trigonometry | Khan Academy

TL;DR

The graphs of y=sin(theta) and y=cos(theta) intersect twice between 0 and 2pi.

Transcript

• [Instructor] We're asked at how many points do the graph of y equals sine of theta and y equal cosine theta intersect for theta between zero and two pi? And it's zero is less than or equal to theta which is less than or equal to two pi, so we're gonna include zero and two pi in the possible values for theta. So to do this, I've set up a little ch... Read More

Key Insights

• 😀 The graphs of y=sin(theta) and y=cos(theta) are periodic and repeat themselves every 2pi.
• 💁 The cosine graph starts at 1, decreases to -1, then increases back to 1, forming a curve that oscillates between these values.
• 💁 The sine graph starts at 0, increases to 1, then decreases back to 0, forming a similar oscillating curve.

Questions & Answers

Q: How many times do the graphs of sine and cosine intersect between 0 and 2pi?

The graphs of sine and cosine intersect twice between 0 and 2pi.

Q: What are the coordinates of the points where the graphs intersect?

The first intersection point is approximately (sqrt(2)/2, sqrt(2)/2), and the second intersection point is approximately (-sqrt(2)/2, -sqrt(2)/2).

Q: What are the values of cosine and sine at theta=0?

At theta=0, cosine is equal to 1 and sine is equal to 0.

Q: What are the values of cosine and sine at theta=pi/2?

At theta=pi/2, cosine is equal to 0 and sine is equal to 1.

Summary & Key Takeaways

• The graph of y=cos(theta) intersects the y-axis at 1 (theta=0) and -1 (theta=pi and 2pi), forming a curve that oscillates between these points.

• The graph of y=sin(theta) intersects the x-axis at 0 (theta=0 and 2pi) and 1 (theta=pi), forming a similar oscillating curve.

• Between 0 and 2pi, the two graphs intersect at two points, approximately between 0 and pi/2 and between pi and 3pi/2.