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?
You now have expressions for AN and NC
Using the law of cosines,
AN^2 = CN^2 + 2^2 - 2*CN*2 cos∠ACN
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To find the constants c and d in the equation cos ∠ACN ≈ c + dx^2, we need to consider the given information and relate it to the equation involving (CN)^2 = 4 - 0.64x^2.
Let's start by analyzing the given triangle and the relationships between the sides and angles.
In triangle ABC:
AC = 2 cm (given),
AB = x cm (variable length),
∠ABC = π/2 rad (given right angle),
∠BAC = ϴ rad (variable angle).
Now, let's consider the point N on AB such that the ratio of AN to AB is 2:5. This means that AN = (2/5)AB and BN = (3/5)AB.
To find CN, we can use the Pythagorean theorem in triangle ACN:
CN^2 = AC^2 + AN^2. (1)
Substituting the values we have:
CN^2 = 2^2 + (2/5)^2 AB^2.
= 4 + (4/25)AB^2.
Now, recall the equation from part 1: x ≈ a + bϴ^2.
To relate x and AB in terms of ϴ, we can use the approximation given in part 1: x ≈ a + bϴ^2.
Substituting AB = x in equation (1) and using the approximation x ≈ a + bϴ^2:
CN^2 = 4 + (4/25)(a + bϴ^2)^2.
Expanding and simplifying:
CN^2 = 4 + (4/25)(a^2 + 2abϴ^2 + b^2ϴ^4).
Comparing this result with (CN)^2 = 4 - 0.64x^2, we can equate the corresponding terms:
∵ 4 - 0.64x^2 = 4 + (4/25)(a^2 + 2abϴ^2 + b^2ϴ^4).
Simplifying further, we get:
-0.64x^2 = (4/25)(a^2 + 2abϴ^2 + b^2ϴ^4).
Dividing both sides by -0.64 and rearranging:
x^2 = -(25/64)(a^2 + 2abϴ^2 + b^2ϴ^4).
Now, we recall the Taylor Series expansion for cos ϴ:
cos ϴ ≈ 1 - (ϴ^2/2) + (ϴ^4/24) - ...
Comparing this with the equation x^2 = -(25/64)(a^2 + 2abϴ^2 + b^2ϴ^4), we can see similarities.
By matching the corresponding terms, we can conclude that:
c = 1 - (a^2/2) - (25b^2/64), and
d = -(25ab/32).
Therefore, the constants c and d in the equation cos ∠ACN ≈ c + dx^2 are determined as c = 1 - (a^2/2) - (25b^2/64) and d = -(25ab/32).
Note: To accurately solve for the constants a and b, further information or restrictions may be required based on the problem or specific context.