Law of sines for missing angle | Trig identities and examples | Trigonometry | Khan Academy

TL;DR

Using the Law of Sines and Law of Cosines, you can calculate the angles of a triangle based on given side lengths.

Transcript

Voiceover:Say you're out flying kites with a friend and right at this moment you're 40 meters away from your friend and you know that the length of the kite's string is 30 meters, and you measure the angle between the kite and the ground where you're standing and you see that it's a 40 degree angle. What you're curious about is whether you can use ... Read More

Key Insights

• 🔺 Trigonometry can be used to solve for unknown angles in triangles.
• 🔺 The Law of Cosines relates the sides and angles of a triangle, but it requires more information than what is given.
• 🥳 The Law of Sines allows us to determine the ratio between the sine of each angle and the length of its opposite side.
• 👯 In the given scenario, the Law of Sines can be used in combination with the distance between two people to calculate the angle between the string and the ground.
• 🎭 Calculators can be used to perform complex trigonometric calculations efficiently.
• 🛝 Rounding to the nearest hundredth of a degree provides a more precise measurement.

Questions & Answers

Q: Why can't the Law of Cosines be used in this situation?

The Law of Cosines requires knowledge of three sides or two sides and the angle between them, which we don't have in this scenario.

Q: How can the Law of Sines be used to solve the problem?

The Law of Sines states that the ratio of the sine of each angle to the length of its opposite side is constant. By knowing the angle at one person's position and the opposite side length, we can calculate the angle between the string and the ground.

Q: What is the equation used to solve for the angle using the Law of Sines?

The equation is sine(theta) / 40 = sin(40) / 30, where theta represents the desired angle.

Q: How do we solve for the angle using the Law of Sines?

To solve for theta, we multiply both sides of the equation by 40, giving us 4/3 sin(40) = sin(theta). Taking the inverse sine of both sides gives us theta, which equals approximately 58.99 degrees.

Q: How can the other angle of the triangle be found?

The other angle can be found by subtracting the known angles from 180 degrees. In this case, it is 180 - 40 - 58.99, which equals approximately 81.01 degrees.

Summary & Key Takeaways

• The problem involves finding the angle between the string of a kite and the ground based on the distance between two people flying the kite and the angle at one person's position.

• The Law of Cosines cannot be applied because not enough information is known about the triangle.

• The Law of Sines can be used to find the angle, but additional information is needed.

• By using the Law of Sines and the known distance between the two people, the angle between the string and the ground can be calculated.