Question

c b a c a b for the right triangle with angles abc and sides abc, write the equations required that define each a) sin a = b) cos a = c) tan a = cos b sin b = d) 1

Answer 1

The equations for a right triangle's sine, cosine, and tangent of angle a are 'sin a = b/c', 'cos a = a/c', and 'tan a = b/a' respectively; 'sin b = cos a = a/c'; and the Pythagorean theorem confirms these relationships with 'a² + b² = c²'.

Step-by-step explanation:

The student is asking about the trigonometric functions of a right triangle where the sides are labeled a (adjacent), b (opposite), and c (hypotenuse), and a is also one of the angles. The equations that define the sine (sin), cosine (cos), and tangent (tan) of angle a would be:

• sin a = b/c
• cos a = a/c
• tan a = b/a

For the cosine of angle b, it would be cos b = a/c. Since the sine of an angle is the cosine of its complement, and a and b are complementary in a right triangle, sin b = cos a = a/c. Finally, to establish the equation '1', we can refer to the Pythagorean theorem which states that in a right triangle a² + b² = c².

Answer 2

The equations for a right triangle's sine, cosine, and tangent of angle a are 'sin a = b/c', 'cos a = a/c', and 'tan a = b/a' respectively; 'sin b = cos a = a/c'; and the Pythagorean theorem confirms these relationships with 'a² + b² = c²'.

Step-by-step explanation:

The student's question pertains to the calculation of trigonometric functions for a right-angled triangle with sides labeled a, b, and c, where c is the hypotenuse, a and b are the lengths of the other two sides, and angles are labeled as ABC with A being the right angle.

• Sine of angle a (sin a) is equal to the length of the opposite side (b) divided by the length of the hypotenuse (c), so sin a = b/c.
• Cosine of angle a (cos a) is equal to the length of the adjacent side (a) divided by the length of the hypotenuse (c), so cos a = a/c.
• Tangent of angle a (tan a) is equal to the sine of angle a divided by the cosine of angle a, which is also equal to the length of the opposite side divided by the length of the adjacent side (b/a), so tan a = b/a.
• The sine of angle b (sin b) is equal to the length of the opposite side (a) divided by the length of the hypotenuse (c), so sin b = a/c.

These equations rely on the Pythagorean theorem, which states that in a right triangle, a² + b² = c². Moreover, the values of sin, cos, and tan functions must be consistent with the principles stated, as these are fundamental to trigonometry and geometry.