Sin Values of 87 and 89 Degrees

  • Context: MHB 
  • Thread starter Thread starter juantheron
  • Start date Start date
  • Tags Tags
    Degrees Sin
Click For Summary
SUMMARY

The value of the product $\sin (1^\circ) \cdot \sin (3^\circ) \cdot \sin (5^\circ) \cdots \sin (87^\circ) \cdot \sin (89^\circ)$ is definitively calculated as \(4.0194366942304562 \times 10^{-14}\). This calculation was discussed among forum members, with one user, Sudhakara, noting the difficulty in solving it algebraically. Alternative methods, including the use of complex numbers and nth roots of unity, were suggested for further exploration.

PREREQUISITES
  • Understanding of trigonometric functions, specifically sine values.
  • Familiarity with algebraic manipulation of trigonometric products.
  • Knowledge of complex numbers and their applications in trigonometry.
  • Basic understanding of roots of unity and their significance in solving equations.
NEXT STEPS
  • Research the properties of trigonometric products, particularly involving sine functions.
  • Explore the application of complex numbers in trigonometric identities.
  • Learn about nth roots of unity and their role in solving polynomial equations.
  • Investigate methods for simplifying products of sine functions using algebraic techniques.
USEFUL FOR

Mathematicians, students studying trigonometry, and anyone interested in advanced algebraic techniques for solving trigonometric equations.

juantheron
Messages
243
Reaction score
1
The value of $\sin (1^0).\sin (3^0).\sin (5^0)...\sin (87^0).\sin (89^0)$

where all angles are in degree
 
Mathematics news on Phys.org
jacks said:
The value of $\sin (1^0).\sin (3^0).\sin (5^0)...\sin (87^0).\sin (89^0)$

where all angles are in degree

Hi jacks,

I haven't found a way solve this algebraically. But if you are interested about the answer it is, \(4.0194366942304562\times 10^{-14}\)
 
Sudharaka said:
Hi jacks,

I haven't found a way solve this algebraically. But if you are interested about the answer it is, \(4.0194366942304562\times 10^{-14}\)

Thanks Sudhakara

I am trying to find it with the help of complex no.(like nth -roots of unity)
 
jacks said:
The value of $\sin (1^0).\sin (3^0).\sin (5^0)...\sin (87^0).\sin (89^0)$

where all angles are in degree
Follow the method used in http://www.mathhelpboards.com/showthread.php?253-Simplify-cos(a)cos(2a)cos(3a)-cos(999a)-if-a-(2pi)-1999&p=1517&viewfull=1#post1517, noting that $x=\pm1^\circ,\pm3^\circ,\pm5^\circ,\ldots,\pm89 ^\circ$ are the solutions of the equation $\cos(90x) = 0.$
 

Similar threads

MHB Evaluate the product of sines: sin1sin2sin3…sin89
  • · Replies 5 ·
Replies
5
Views
3K
MHB Finding x in Degrees: Solving a Quadratic Equation for Trigonometric Functions
  • · Replies 5 ·
Replies
5
Views
1K
Undergrad Transforming coordinates between Cartesian and spherical
  • · Replies 1 ·
Replies
1
Views
2K
MHB Prove Triangle Inequality: $\sqrt{2}\sin A-2\sin B+\sin C=0$
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Equation to graph a sine wave that acts like a point on a unit circle
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad A Trick to Memorizing Trig Special Angle Values Table
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Determine ## \beta ## as a function of ##\theta## linkage
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Trig Manipulations I'm Not Getting
  • · Replies 1 ·
Replies
1
Views
1K
MHB Find Coefficient of Friction for Toy Car Rolling Down Slope
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad How do I do this calculation involving the SIN function?
  • · Replies 9 ·
Replies
9
Views
2K