Law of Sines - Solving Oblique Triangle

Summary

In this educational video, Smithy Turgon introduces the Law of Sines, a fundamental principle used for solving oblique triangles. He demonstrates the application of the Law of Sines formula by working through an example involving a triangle with given angles and a side length. The video guides viewers step-by-step to find the missing angles and sides, using a calculator to perform the necessary trigonometric calculations. Smithy Turgon emphasizes the importance of understanding the relationship between the angles and sides of a triangle, ultimately providing the measurements for all sides and angles in the example.

Takeaways

• 📚 The video discusses the Law of Sines, a mathematical principle used to solve oblique triangles.
• 🔍 The Law of Sines formula is presented as a ratio of the sides of a triangle to the sines of their opposite angles: a/sin(A) = b/sin(B) = c/sin(C).
• 📐 The script introduces an alternative version of the Law of Sines formula with sine functions in the numerator and sides in the denominator.
• 📈 The video provides an example problem involving a triangle with given angles B and C, and side a, aiming to find angle A and sides b and c.
• 🧭 It explains that the sum of angles in a triangle is 180 degrees, and uses this to find the missing angle A.
• 🔢 The script demonstrates the use of a calculator to solve for side B using the Law of Sines formula.
• 📝 Cross-multiplication is used to isolate the unknown side B in the ratio, and the sine of the known angle is used to find its length.
• 📉 The process is repeated to find side C, again using the Law of Sines and cross-multiplication.
• 📊 The video emphasizes the importance of accurate calculations and rounding off to the appropriate number of decimal places.
• 📢 The presenter, Smithy Turgon, encourages viewers to follow him on social media and subscribe to his channel for more educational content.
• 👋 The video concludes with a reminder of the presenter's name and a friendly sign-off.

Q & A

What is the law of sines used for?

The law of sines is used for solving oblique triangles, particularly when you have certain angles and sides and need to find the missing parts.

What is the formula for the law of sines?

The formula for the law of sines is \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \), where lower case letters represent the sides of the triangle and the corresponding upper case letters represent the angles opposite those sides.

In the given example, what are the known values in triangle ABC?

In the example, angle B is 141 degrees, angle C is 23 degrees, and the length of side a is 9 units.

How is the missing angle A calculated in the example?

Angle A is calculated using the formula \( 180^\circ - \text{Angle B} - \text{Angle C} \), which in this case is \( 180^\circ - 141^\circ - 23^\circ = 16^\circ \).

What is the first step to find the length of side B using the law of sines?

The first step is to use the formula \( \frac{a}{\sin A} = \frac{b}{\sin B} \) and cross-multiply to solve for side B.

How is the length of side B found in the example?

By cross-multiplying \( 9 \sin 141^\circ \) with \( \sin 16^\circ \) and dividing, the length of side B is approximately 20.5 units.

What formula is used to find the length of side C?

The same law of sines formula is used, but with the known values for side a and angle C to find the length of side C.

How is the length of side C calculated in the example?

By cross-multiplying \( 9 \sin 23^\circ \) with \( \sin 16^\circ \) and dividing, the length of side C is approximately 12.8 units.

What is the significance of the law of sines in solving for the missing sides of a triangle?

The law of sines allows you to relate the ratios of the sides of a triangle to the sines of their opposite angles, which is crucial when you have some angles and sides known and need to find the others.

What is the final step in the example after finding the missing angles and sides?

The final step is to verify that all the angles add up to 180 degrees and that the sides satisfy the law of sines, ensuring the solution is correct.