Pythagorean trig identity from soh cah toa | Trigonometry | Khan Academy

TL;DR

Explore the relationship between trigonometric functions and the Pythagorean theorem, and discover the most important trig identity: sine squared theta plus cosine squared theta equals 1.

Transcript

So we've got a right triangle drawn over here where this base's length is a, the height here is b, and the length of the hypotenuse is c. And we already know when we see something like this, we know from the Pythagorean theorem, the relationship between a, b, and c, we know there's a squared plus b squared is going to be equal to the hypotenuse squ... Read More

Key Insights

• 🙃 The Pythagorean theorem, which relates the sides of a right triangle, can be connected to trigonometric functions.
• 🥳 Sine represents the ratio of the length of the opposite side to the hypotenuse, while cosine represents the ratio of the length of the adjacent side to the hypotenuse.
• ❎ By squaring sine and cosine, it is possible to derive the trig identity of sine squared theta plus cosine squared theta equals 1.
• 👻 This trig identity is powerful and allows for solving equations involving trigonometric functions.
• ❓ The identity provides a fundamental connection between trigonometry and geometry.
• 🔨 The sohcahtoa definition of trig functions is a useful tool for understanding their relationships to the Pythagorean theorem.

Questions & Answers

Q: How can trigonometric functions be related to the Pythagorean theorem?

Trigonometric functions, like sine and cosine, can be related to the Pythagorean theorem by considering their definitions in terms of the sides of a right triangle. This allows us to explore the connection between trig functions and the lengths of the triangle sides.

Q: What is the trig identity that relates sine and cosine?

The most important trig identity is sine squared theta plus cosine squared theta equals 1. This identity shows that the sum of the squares of sine and cosine for the same angle will always equal 1. It is a fundamental relationship in trigonometry.

Q: Why is the sine squared plus cosine squared identity important?

This trig identity is significant because it allows us to solve for unknown angles or trig function values. If we know the sine of an angle, for example, we can use this identity to find the cosine of that angle and vice versa. It provides a powerful tool for solving trigonometric equations.

Q: What is the motivation for the unit circle definition of trig functions?

While this video focuses on the sohcahtoa definition of trig functions (sine, cosine, and tangent), the unit circle definition is another approach to understanding trigonometry. The unit circle definition builds on the relationships between angles, coordinates on the unit circle, and trig functions, providing a deeper understanding of their properties.

Summary & Key Takeaways

• The Pythagorean theorem states that in a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse.

• Trigonometric functions, such as sine and cosine, can be related to the Pythagorean theorem through their definitions.

• By squaring sine and cosine, it is possible to combine them to obtain the trig identity: sine squared theta plus cosine squared theta equals 1.