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What is the Trigonometric ratio for sine Z

What is the Trigonometric ratio for sine Z — Explanation & Interactive Diagram

What is the Trigonometric ratio for sine Z? In this section we explain the definition, show a live diagram you can interact with, provide examples, and answer common questions — all focused on the phrase What is the Trigonometric ratio for sine Z.

Definition — What is the Trigonometric ratio for sine Z?

By definition, the trigonometric ratio for sine Z (written sin Z) is the ratio of the length of the side opposite angle Z to the hypotenuse in a right triangle:

sin Z = opposite / hypotenuse

On the unit circle (radius = 1), sin Z equals the y-coordinate of the point reached by rotating from the positive x-axis by angle Z (measured in degrees or radians).

Unit circle diagram (interactive)

Move the slider to change Z (angle) and watch the point move — the y-value displayed is sin Z, illustrating What is the Trigonometric ratio for sine Z.

sin Z = 0.5
cos Z (x) = 0.866

Description of diagram

This interactive shows the unit circle (radius = 1). The point on the circumference corresponds to angle Z. The vertical projection from the point to the x-axis gives the sin Z value (y-coordinate). Use the slider to change the angle and see how What is the Trigonometric ratio for sine Z varies.

Quick notes

  • Range: −1 ≤ sin Z ≤ 1
  • Period: sin(Z + 360°) = sin Z (or sin(z + 2π) = sin z)
  • Angle units: degrees or radians

Properties & formulas (using the keyword)

Basic identities including “What is the Trigonometric ratio for sine Z”

  • sin²Z + cos²Z = 1
  • sin(-Z) = -sin Z
  • sin(90° − Z) = cos Z
  • sin(A ± B) = sin A cos B ± cos A sin B

Worked examples — answering “What is the Trigonometric ratio for sine Z”

  1. Right triangle: If the opposite = 3 and hypotenuse = 5, sin Z = 3/5 = 0.6.
  2. Unit circle: Z = 30° → sin 30° = 0.5.
  3. Using identities: If cos Z = 0.8 and Z in first quadrant, sin Z = sqrt(1 - 0.8²) = 0.6.

Frequently asked questions — What is the Trigonometric ratio for sine Z (20)

1. What is the trigonometric ratio for sine Z in words?
Sin Z is the ratio of the side opposite angle Z to the hypotenuse (in a right triangle), or the y-coordinate on the unit circle.
2. How do I calculate sin Z from a right triangle?
Divide the length of the opposite side by the hypotenuse: sin Z = opposite / hypotenuse.
3. What is sin 90° for Z = 90°?
sin 90° = 1.
4. Can sin Z be negative?
Yes — sin Z is negative when Z is in quadrants III and IV (angles between 180°–360°).
5. What units can Z be in?
Z can be in degrees or radians (e.g., 180° = π radians).
6. What are common exact values of sin Z?
sin 0°=0, sin 30°=1/2, sin 45°=√2/2, sin 60°=√3/2, sin 90°=1.
7. How does the unit circle show “What is the Trigonometric ratio for sine Z”?
On the unit circle, sin Z is the vertical coordinate (y-value) of the point at angle Z.
8. Is sin Z periodic?
Yes, sin Z repeats every 360° (2π radians): sin(Z + 360°) = sin Z.
9. How to compute sin Z on a calculator?
Set the correct unit (deg/rad), enter the angle Z, then press the sin function.
10. What is the derivative of sin Z?
d/dZ [sin Z] = cos Z (using Z in radians).
11. What is the integral of sin Z?
∫ sin Z dZ = −cos Z + C (for Z in radians).
12. How to find angle Z if sin Z is known?
Use the inverse sine (arcsin) function: Z = arcsin(value), paying attention to the correct quadrant.
13. Can sin Z = 2?
No — sine values are bounded between −1 and 1.
14. Relationship between sin Z and cos Z?
sin²Z + cos²Z = 1 always holds.
15. What is sin(180° − Z)?
sin(180° − Z) = sin Z (sine is symmetric about 90°).
16. Common mistakes when using sin Z?
Using degrees vs radians incorrectly and misidentifying which side is opposite or hypotenuse in triangle problems.
17. How to express sin Z in terms of complex exponentials?
sin Z = (e^{iZ} − e^{−iZ}) / (2i), when Z is in radians.
18. How is sin Z used in real-world contexts?
Wave motion, signal processing, engineering, navigation, and any periodic phenomena use sine functions.
19. Can you compute sin Z from coordinates?
Yes: for a point (x,y) on the unit circle at angle Z, sin Z = y.
20. Where else should I learn about sin Z?
Good resources include math textbooks, Khan Academy, and university lecture notes (links below).
Diagram explaining What is the Trigonometric ratio for sine Z.

Resources links

Internal page: Explore Detailed study of Trigonometry (internal)
Outbound resource: Khan Academy — Trigonometry

Abdul Laskar

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Abdul Laskar
2 months ago

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