Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

0. Functions

Trigonometric Identities

Calculus

0. Functions

Trigonometric Identities

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2:44 minutes

Problem 63

Textbook Question

A triangle has side c = 2 and angles A = π/4 and B = π/3. Find the length a of the side opposite A.

Verified step by step guidance

First, recognize that you are dealing with a triangle where you know two angles and one side. This is a case for the Law of Sines, which relates the sides of a triangle to the sines of its angles.

The Law of Sines states: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). In this problem, you need to find side 'a', and you know side 'c' and angles A and B.

Calculate the third angle C using the fact that the sum of angles in a triangle is \( \pi \) radians. So, \( C = \pi - A - B \). Substitute \( A = \frac{\pi}{4} \) and \( B = \frac{\pi}{3} \) to find \( C \).

Now, apply the Law of Sines: \( \frac{a}{\sin A} = \frac{c}{\sin C} \). Substitute the known values: \( c = 2 \), \( A = \frac{\pi}{4} \), and the calculated \( C \).

Solve for 'a' by rearranging the equation: \( a = c \cdot \frac{\sin A}{\sin C} \). Substitute the values and calculate the sine of the angles to find the length of side 'a'.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Law of Sines

The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all three sides and angles. This relationship can be expressed as a/sin(A) = b/sin(B) = c/sin(C). In this problem, it allows us to find the unknown side 'a' by relating it to the known side 'c' and the angles A and B.

Angle Measures in Radians

In calculus and trigonometry, angles can be measured in degrees or radians. Radians are a more natural unit for mathematical analysis, where π radians equals 180 degrees. Understanding how to convert between these two systems is crucial, especially since the angles given in the problem are in radians, which will be used in the sine function calculations.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the ratios of its sides. The sine function, in particular, is essential for solving triangles, as it provides a way to calculate unknown side lengths when angles are known. In this problem, we will use the sine of angles A and B to find the length of side 'a' opposite angle A.