Sine of non special angle | Trig identities and examples | Trigonometry | Khan Academy

TL;DR

The video explains how to find the sine of seven pi over 12 without using a calculator.

Transcript

Voiceover:What I want to attempt to do in this video is figure out what the sine of seven pi over 12 is without using a calculator. And so let's just visualize seven pi over 12 in the unit circle. One side of the angle is going along the positive x-axis and then let's see, if we go straight up, that's pi over two, which is the same thing as six pi ... Read More

Key Insights

• 🔨 The unit circle is a useful tool in visualizing angles and calculating trigonometric functions.
• 🔺 Special right triangles, like the 30-60-90 and 45-45-90 triangles, have well-known ratios that can aid in finding trigonometric values.
• 🔺 Decomposing an angle into combinations of known angles allows us to use trigonometric formulas effectively.
• 🔺 The angle addition formula for sine can help simplify and express complex angles as sums of known angles.
• 🤨 The sine of seven pi over 12 can be found by relating it to the sine and cosine of pi over four and pi over three, respectively.

Questions & Answers

Q: How is the sine of an angle defined in terms of the unit circle?

The sine of an angle in the unit circle is the y-coordinate of the point where the ray intersects the circle. It represents the length of a line perpendicular to the x-axis.

Q: Why is the triangle with an angle of seven pi over 12 relevant in finding the sine?

The triangle is created to represent the given angle and is used to relate the sides to trigonometric functions. It helps in finding the sine by analyzing the lengths of its sides.

Q: Can the sine of seven pi over 12 be directly calculated using known trigonometric values?

No, the sine of seven pi over 12 cannot be directly calculated. It requires decomposing the angle into combinations of known angles and applying trigonometric formulas to find the solution.

Q: What is the angle addition formula for sine in trigonometry?

The angle addition formula states that the sine of the sum of two angles is equal to the product of the sines of the angles, added to the product of the cosines of the angles.

Summary & Key Takeaways

• The video demonstrates how to visualize and locate the angle seven pi over 12 on the unit circle.

• By relating the angle to known trigonometric values of special right triangles (30-60-90 and 45-45-90), the video presents a method to find the sine and cosine of seven pi over 12.

• Through angle addition formula for sine, the video simplifies and calculates the sine of seven pi over 12 as the square root of two plus square root of six, divided by four.