Law of cosines to determine grade | Trig identities and examples | Trigonometry | Khan Academy

TL;DR

Trigonometry can be used to calculate the steepness of a slope or incline by using the law of cosines.

Transcript

Voiceover:Let's say you're studying some type of a little hill or rock formation right over here. And you're able to figure out the dimensions. You know that from this point to this point along the base, straight along level ground, is 60 meters. You know the steeper side, steeper I guess surface or edge of this cliff or whatever you wanna call it,... Read More

Key Insights

• 🔨 Trigonometry can be a useful tool in calculating the steepness or inclination of a slope or incline.
• 🔺 The law of cosines allows for the calculation of an unknown angle in a triangle when the lengths of the sides are known.
• 🔺 By applying the law of cosines to a triangle representing a slope, the angle of inclination can be determined.
• 🏑 Trigonometry is widely used in various fields, including engineering, architecture, and physics, to analyze and design slopes and inclines.
• 🔺 The law of cosines provides a mathematical formula for calculating angles and distances in non-right triangles.
• 🔺 Trigonometry helps in understanding the relationship between angles and sides of a triangle, enabling accurate measurements and calculations.
• 👷 Trigonometric calculations can assist in determining the safety and stability of slopes, especially in construction and geotechnical engineering.

Questions & Answers

Q: How can trigonometry be used to determine the steepness of a slope?

Trigonometry can be used by applying the law of cosines to a triangle representing the slope. By knowing the lengths of the sides of the triangle, the angle of inclination can be calculated.

Q: What is the law of cosines and how does it relate to trigonometry?

The law of cosines states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of the lengths of those sides and the cosine of the included angle.

Q: What are the dimensions needed to calculate the steepness of a slope using trigonometry?

Trigonometry requires knowing three dimensions of the slope: the length of the base, the length of the steeper side, and the length of the less steep side of the triangle representing the slope.

Q: How can the angle of inclination be determined using trigonometry?

By applying the law of cosines to a triangle representing the slope, the unknown angle of inclination can be calculated by rearranging the equation and solving for the angle.

Summary & Key Takeaways

• Trigonometry can be used to determine the steepness or inclination of a slope or incline.

• The law of cosines can be applied when three sides of a triangle are known and an angle needs to be calculated.

• By applying the law of cosines to a triangle representing a slope, the angle of inclination can be determined.