# sines

1. ### Problem #20: Sum of Sines

We haven't done a Trigonometry problem in a while, so here we go! Evaluate: sin^4 \left ( \frac{\pi}{8} \right ) + sin ^4 \left ( \frac{3 \pi}{8} \right ) + sin ^4 \left ( \frac{5 \pi}{8} \right ) + sin ^4 \left ( \frac{7 \pi}{8} \right ) -Dan
2. ### reciprocal form Of law of sines . question

Hello i am a bit confused when it comes to this law. How do i know when the reciprocal form is to be used And how do i know when its not to be used? If anyone could make an example. Thanks
3. ### Applying the Laws of Sines or Cosines

Solve the triangle. A = 19°, C = 102°, c = 6 choices are B = 31°, a ≈ 18, b ≈ 15.8 B = 59°, a ≈ 2, b ≈ 5.3 B = 59°, a ≈ 18, b ≈ 15.8 B = 59°, a ≈ 18, b ≈ 5.3
4. ### Law of Sines

Two triangles can be formed with the given information. Use the Law of Sines to solve the triangles. B = 49°, a = 16, b = 14 Choices are A = 30.4°, C = 100.6°, c = 18.2; A = 149.6°, C = 79.4°, c = 18.2 A = 30.4°, C = 100.6°, c = 10.7; A = 149.6°, C = 79.4°, c = 10.7 A = 59.6°, C = 71.4°...
5. ### Applying the Laws of Sines or Cosines

Solve the triangle. B = 36°, a = 41, c = 17 choices are b ≈ 29, C ≈ 20, A ≈ 124 b ≈ 44, C ≈ 20, A ≈ 124 b ≈ 29, C ≈ 121, A ≈ 23 b ≈ 44, C ≈ 24, A ≈ 120
6. ### Law Of Sines

Using Law of Sines for the ambiguous case: SSA : Given the triangle with the following sides: A = 25 degrees, a = 8 cm, and b = 15cm , we begin by drawing a "vague triangular diagram": Step 1: Find the value of h ( this determines how many triangles you will have) : Step 2: Draw in the side...
7. ### Law of Sines

The bearing from the Pine Knob fire tower to the Colt Station fire tower is N 65 degrees East, and the two towers are 30 km apart. A fire spotted by rangers in each tower has a bearing of N 80 degrees East from Pine Knob and S 70 degrees E from Colt Station. Find the distance of the fire from...
8. ### Use the law of sines to deduce the law of tangent

Use the law of sines to deduce the law of tangents for triangle ABC: (a-b)/(a-b) = (tan(1/2)(A-B))/(tan(1/2)(A-B)) I've put a lot of thought into this homework problem and I hardly even know where to begin. I've attended every class and done all of the homework and none of this seems familiar...
9. ### Sum of Product of Sines

Hello, I'm trying to show that \sum _{j=1}^{N-1} \sin \left(\frac{j \pi y}{N}\right)\sin\left(\frac{j \pi k}{N}\right)=0, if y does not equal k (with y and k both being integers). I've tried using the trig identity for the product of sines. That gives \sum _{j=1}^{N-1} \left [\cos\left (\frac{j...
10. ### Solving Vector Triangles (Law of Sines and Cosines possibly)

Hey all, I really don't know where to begin this problem. Three vectors, Vector AO is 400 in magnitude and direction going in -x from point A to O. (or can rearrange I believe to make another triangle piece) Vector AB is unknown going from point A to B. Vector AC is 525 in magnitude going from...
11. ### complex algebra? series sum of sines or cosines

how do i calculate the sum of each of these series? sin(A) + sin(A+B) + sin(A+2B) + sin(A+3B) .... + sin(A+(n-1)B) cos(A) + cos(A+B) + cos(A+2B) + cos(A+3B) .... + cos(A+(n-1)B) please show step by step how to do it, not just the answer.
12. ### Surveying Property (Law of Sines)

can someone help me sketch out this? I tried by myself but always end up with the wrong answer.
13. ### Law of Sines Application Help

A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 5 mi. apart, to be 32 degrees and 48 degrees as shown in the figure. a) Find the distance of the plane from point A. b) Find the elevation of the plane. imgur: the simple image sharer...
14. ### Formula product of sines

Does anyone know how to prove this formula? 2^{n-1}\prod_{k=0}^{n-1}\sin(x+\frac{k\pi}{n}) = \sin(nx) All suggestions, ideas are welcome. Below you find my unsuccessfull attempt using complex numbers. When you convert it to complex numbers the equality can be rewritten as...
15. ### The product of sines and the product of cosines of (k\pi)/(2n+1) for 0<k<n+1

I have to prove the following identities: \prod_{k=1}^n \cos\frac{k\pi}{2n+1}=\frac{1}{2^n} \prod_{k=1}^n \cos\frac{k\pi}{2n+1}=\frac{\sqrt{2n+1}}{2^n} The suggestion is to prove and use this: \sum_{k=0}^{2n}z^k=\prod_{k=1}^n \left (1-2z\cos\frac{2k\pi}{2n+1}+z^2\right ) for z\in\mathbb{C}...
16. ### Another proof with Law of Sines

Let p and q be the radii of two circles through A (with A being a point on Triangle ABC), touching BC at B and C, respectively. Prove the pq = R squared. (R is the radius of the circle circumscribed around ABC). I have been trying over and over to do this problem, and I just can't get it.
17. ### Proof using Law of Sines

For any triangle ABC, even if B or C is an obtuse angle, a = b cos C + c cos B. Use the Law of Sines to deduce the "addition formula" sin (B + C) = sin B cos C + sin C cos B I got most of it but can't figure out the end. I replaced b with 2R sin B and c with 2R sin C so a = 2R sin B...
18. ### Proof using Law of Sines

For any triangle ABC, even if B or C is an obtuse angle, a = b cos C + c cos B. Use the Law of Sines to deduce the "addition formula" sin (B + C) = sin B cos C + sin C cos B I got most of it but can't figure out the end. I replaced b with 2R sin B and c with 2R sin C so a = 2R sin B...
19. ### the zero of the third derivative of iterated sines

Let f_n\colon[0,\pi/2]\longrightarrow[0,1] for n=0,1,... and \begin{cases}f_0=\sin\\f_{n+1}=\sin\circ f_n\end{cases} I would like to prove that for n=1,2,... the third derivative f_n^{(3)} has a unique zero in (0,\pi/2). I thought calculating the third derivative and solving an equation...
20. ### Ambiguous case using law of sines

Hello, here is the data I am given to work with: Angle A=30 deg Side a=3 Angle B=unknown Side b=unknown Angle C=unknown Side c=4 So I know that I can take the 3/sin30 = 4/sinC Rearranging that, I am left with SinC = Sin(30)(4)/3 which can further be simplified to 41.81 deg. So does having...