Ch. 1 - Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 1 - Functions

Problem 63

Chapter 1, Problem 63

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.

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Related Practice

Textbook Question

Two wires stretch from the top T of a vertical pole to points B and C on the ground, where C is 10 m closer to the base of the pole than is B. If wire BT makes an angle of 35° with the horizontal and wire CT makes an angle of 50° with the horizontal, how high is the pole?

Textbook Question

In Exercises 65–68, ABC is a right triangle with the right angle at C. The sides opposite angles A, B, and C are a, b, and c, respectively.

a. Find a and b if c = 2, B = π/3.

b. Find a and c if b = 2, B = π/3.

Textbook Question

Apply the formula for cos (A − B) to the identity sin θ = cos (π/2 − θ) to obtain the addition formula for sin (A + B).

Textbook Question

The law of sines The law of sines says that if a, b, and c are the sides opposite the angles A, B, and C in a triangle, then

(sin A) / a = (sin B) / b = (sin C) / c

Use the accompanying figures and the identity sin (π − θ) = sin θ, if required, to derive the law.

Textbook Question

In Exercises 65–68, ABC is a right triangle with the right angle at C. The sides opposite angles A, B, and C are a, b, and c, respectively.

a. Express sin A in terms of a and c.

b. Express sin A in terms of b and c.

Textbook Question

In Exercises 59–62, sketch the graph of the given function. What is the period of the function?

𝔂 = cos πx/2