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

Introduction to Trigonometric Functions

Calculus

0. Functions

Introduction to Trigonometric Functions

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2:15 minutes

Problem 68

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.

Verified step by step guidance

Step 1: Understand the problem setup. We have a right triangle ABC with the right angle at C. The sides opposite angles A, B, and C are labeled as a, b, and c, respectively. In a right triangle, the side opposite the right angle is the hypotenuse, which is c in this case.

Step 2: Recall the definition of the sine function in a right triangle. The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

Step 3: To express sin A in terms of a and c, use the definition of sine. Since angle A is opposite side a and the hypotenuse is c, we have:

\sin A = \frac{a}{c}

Step 4: To express sin A in terms of b and c, we need to use the Pythagorean identity. In a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). Therefore,

c^{2} = a^{2} + b^{2}

. Solve for a:

a = \sqrt{c^{2} - b^{2}}

Step 5: Substitute the expression for a from Step 4 into the sine formula from Step 3 to express sin A in terms of b and c:

\sin A = \frac{\sqrt{c^{2} - b^{2}}}{c}

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Key Concepts

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

Sine Function

The sine function is a fundamental trigonometric function defined for a right triangle as the ratio of the length of the side opposite an angle to the length of the hypotenuse. For angle A in triangle ABC, sin A = opposite/hypotenuse = a/c, where 'a' is the side opposite angle A and 'c' is the hypotenuse.

Right Triangle Properties

In a right triangle, the relationship between the angles and sides is governed by the Pythagorean theorem and trigonometric ratios. The right angle (90 degrees) allows for the use of sine, cosine, and tangent to relate the angles to the lengths of the sides, which is essential for solving problems involving right triangles.

Trigonometric Ratios

Trigonometric ratios are the ratios of the lengths of sides of a right triangle relative to its angles. These ratios (sine, cosine, tangent) are used to express relationships between the angles and sides, allowing for the calculation of unknown lengths or angles in triangle ABC, particularly in expressing sin A in terms of different sides.

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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.

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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.

Key Concepts

Sine Function

Right Triangle Properties

Trigonometric Ratios

Watch next

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.