Question

Which of these is equivalent to the ratio ?

a. sin a
b. cos e
c. tan 8
d. cot

Answer 1

The question appears to be incomplete as it asks for an equivalent trigonometric ratio without specifying the original ratio. Given the context, it involves understanding and applying trigonometric identities and the laws associated with triangles, such as the Law of Sines and the Law of Cosines. Moreover, a related concept from another example refers to Snell's Law in Physics.

Step-by-step explanation:

The question seems to ask for an equivalent trigonometric ratio but the actual ratio in question is missing from the prompt. However, based on the provided information, we can deduce that:

1. Sin (a ± β) = sin a cos β ± cos a sin β
2. Cos (α ± β) = cos a cos β - sin a sin β
3. Tan (a ± β) = (tan a + tan B) / (1 - tan a tan B)

Also, if we are looking for trigonometric identities related to the ratios mentioned in options a, b, c, and d, we can reference the double angle formulas and sum-to-product formulas provided in items 13-16:

• Sin 2θ = 2sin θ cos θ
• Cos 2θ = cos² θ - sin² θ
• Sin a + sin β = 2 sin((a + B)/2) cos((a − B)/2)
• Cos a + cos β = 2 cos((a + B)/2) cos((a − B)/2)

For the triangle laws:

• The Law of Sines is a/sin a = b/sin β = c/sin y
• The Law of Cosines is c² = a² + b² - 2ab cos y

Lastly, Check Your Understanding question 8 refers to Snell's Law, which is expressed as n₁ sin d₁ = n₂ sin d₂.