In the triangle ABC, AC=2cm, AB=x cm and angle ABC=π/2 rad.
1. Given that angle BAC=ϴ rad, and that ϴ is a sufficiently small angle, show that x≈ a + bϴ^2, where a and b are constants to be determined.
2. For this part, ϴ is not assumed to be a small angle. The point N lies on AB such that the ratio of AN to AB is 2:5. By showing that (CN)^2= 4 - 0.64x^2, find the constants c and d such that when x is small, cos ∠ACN≈ c + dx^2.
I've done Q1 and half of Q2. I've shown the equation but can't figure out how to find constants c and d. I tried using cosine rule but couldn't get an answer.. Can someone help me?
To find the constants c and d in part 2 of the question, we need to use the cosine rule, as you have correctly mentioned. Let's go through the steps together.
First, let's draw the triangle ABC and point N on AB as described. We know that AN is 2/5 of AB, so let's label AN as (2/5)x and BN as (3/5)x.
Now, let's consider triangle ACN. According to the cosine rule, we have:
(CN)^2 = (AC)^2 + (AN)^2 - 2 * AC * AN * cos(∠ACN).
Substituting the given values, we get:
(CN)^2 = 2^2 + (2/5x)^2 - 2 * 2 * (2/5x) * cos(∠ACN).
Simplifying further, we have:
(CN)^2 = 4 + 4/25x^2 - (8/5x) * cos(∠ACN).
Now, let's focus on finding the value of cos(∠ACN). Since we already know that angle ABC is a right angle (π/2 rad), we can use the right triangle trigonometry identity:
cos(∠ACN) = AN / AC.
Substituting the values, we have:
cos(∠ACN) = (2/5x) / 2 = 1/5x.
Now, let's substitute this back into our previous equation:
(CN)^2 = 4 + 4/25x^2 - (8/5x) * (1/5x).
Simplifying further, we have:
(CN)^2 = 4 + 4/25x^2 - 8/25.
Combining like terms, we get:
(CN)^2 = 100/25 + 4/25x^2 - 8/25.
Simplifying again, we have:
(CN)^2 = (96 + 4x^2)/25.
Comparing this equation to the given equation, we can see that c = 96/25 and d = 4/25.
Therefore, when x is small, cos ∠ACN≈ (96/25) + (4/25)x^2.
I hope this explanation helps you understand how to find the constants c and d using the cosine rule.