BeginnerTopic 15 of 104Engineering Mathematics
Unit circle and identities
sin²θ+cos²θ=1 and related identities simplify oscillation and rotation problems.
In depth
On the unit circle the point at angle θ is (cos θ, sin θ). This gives signs in each quadrant and periodicity of 2π. Identities include sin²+cos²=1, double angle sin2θ=2sinθcosθ, and sum formulas used to combine waveforms.
Key points
- cos is even, sin is odd
- sin(θ+π/2)=cosθ
- Product-to-sum formulas help in modulation analysis
Examplecos2θ = 1-2sin²θ, used to derive average power of sine waves.
Trigonometric identities simplify expressions for oscillation, rotation and wave interference, turning messy sums into single terms.
What is Unit circle and identities?
sin²θ+cos²θ=1 and related identities simplify oscillation and rotation problems.
Why is it important to understand unit circle and identities?
Trigonometric identities simplify expressions for oscillation, rotation and wave interference, turning messy sums into single terms.
Which identity should I memorise first?
sin²θ + cos²θ = 1 is the most useful. Most others can be derived from it or from the angle-addition formulas.
Can an identity help me solve a real circuit?
Yes. Combining sinusoidal voltages with the same frequency into a single phasor or amplitude is an identity-based simplification used in AC analysis.
How is Unit circle and identities used in practice?
Typical applications include combining two sinusoids of the same frequency into one wave, simplifying the rotation of a robot joint in kinematics and reducing a product of sines in a Fourier analysis.
What should I learn before Unit circle and identities?
Study "Radians and angular measure" first (topic 14 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Unit circle and identities?
cos is even, sin is odd. sin(θ+π/2)=cosθ. Product-to-sum formulas help in modulation analysis.
Can you give an example of Unit circle and identities?
cos2θ = 1-2sin²θ, used to derive average power of sine waves.
What comes after Unit circle and identities?
The next topic is "Law of sines and cosines", which builds directly on this one.
How difficult is Unit circle and identities?
It is classed as Beginner in Engineering Mathematics (topic 15 of 104). It is an entry-level topic. It relies only on basic school mathematics and the ideas introduced earlier in this discipline.