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

1. 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.
2. 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.
3. 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: <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>sin</mi><mo>&#xA0;</mo><mi>A</mi><mo>=</mo><mfrac><mi>a</mi><mi>c</mi></mfrac></mrow></math>
4. 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, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi>c</mi><mn>2</mn></msup><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></math>. Solve for a: <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi><mo>=</mo><msqrt><mrow><msup><mi>c</mi><mn>2</mn></msup><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></msqrt></mrow></math>
5. 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: <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>sin</mi><mo>&#xA0;</mo><mi>A</mi><mo>=</mo><mfrac><msqrt><mrow><msup><mi>c</mi><mn>2</mn></msup><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></msqrt><mi>c</mi></mfrac></mrow></math>

Key Concepts

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.