Calcula en cada caso las restantes razones trigonométricas de un angulo agudo si se conoce que:

A) Sen a = raiz de 3/5

B) cos a = 1/3

C) tan a = raiz de 5

D) cos a = 4/5

E) tan a = 5

F) cos a = 0,8

A) cos a = (√22) / 5; tan a = (√66) / 22

B) sin a = (2√2) / 3; tan a = 2√2

C) sin a = (√30) / 6; cos a = (√6) / 6

D) sin a = 3/5; tan a = 3/4

E) sin a = (5√26) / 26; cos a = (√26) / 26

F) sin a = 3/5; tan a = 3/4

Explanation

The ratio for sine is opposite/hypotenuse. This means the side opposite the angle is √3 and the hypotenuse is 5. Using the Pythagorean theorem to find the adjacent side,

(√3) ² + A² = 5²

3+A² = 25

A² = 22

A=√22

This means that cos a = adjacent/hypotenuse = (√22) / 5 and tan a = opposite/adjacent = (√3) / (√22) = (√66) / 22.

B) The ratio for cosine is adjacent/hypotenuse; this means the side adjacent to the angle is 1 and the hypotenuse is 3. Using the Pythagorean theorem to find the side opposite the angle (p),

1² + p² = 3²

1+p² = 9

p² = 8

p=√8 = 2√2

This means that sin a = opposite/hypotenuse = (2√2) / 3 and tan a = opposite/adjacent = (2√2) / 1 = 2√2.

C) The ratio for tangent is opposite/adjacent; this means that the side opposite the angle is √5 and the side adjacent the angle is 1. Using the Pythagorean theorem to find the hypotenuse,

(√5) ²+1² = H²

5+1=H²

6=H²

√6 = H

This means that sin a = opposite/hypotenuse = (√5) / (√6) = (√30) / 6 and cos a = adjacent/hypotenuse = 1 / (√6) = (√6) / 6.

D) The ratio for cosine is adjacent/hypotenuse; this means that the side adjacent the angle is 4 and the hypotenuse is 5. Using the Pythagorean theorem to find the side opposite the angle, p:

4²+p²=5²

16+p²=25

p²=9

p=3

This means that sin a = opposite/hypotenuse = 3/5 and tan a = opposite/adjacent = 3/4.

E) The ratio for tangent is opposite/adjacent;; this means that the side opposite the angle is 5 and the side adjacent the angle is 1. Using the Pythagorean theorem to find the hypotenuse,

5²+1²=H²

25+1=H²

26=H²

√26 = H

This means that sin a = opposite/hypotenuse = 5 / (√26) = (5√26) / 26 and cos a = adjacent/hypotenuse = 1 / (√26) = √26/26.

F) 0.8 = 8/10; The ratio for cosine is adjacent/hypotenuse. This means that the side adjacent the angle is 8 and the hypotenuse is 10. Using the Pythagorean theorem to find the side opposite the angle, p:

8²+p² = 10²

64+p² = 100

p² = 36

p=6

This means that sin a = opposite/hypotenuse = 6/10 = 3/5 and tan a = opposite/adjacent = 6/8 = 3/4.