Sine is the trigonometric ratio that compares the side opposite an angle with the hypotenuse of a right triangle: sin θ = oppositehypotenuse. On the unit circle, sin θ is the y-coordinate of the point at angle θ, so sine values always stay between −1 and 1.
Sine is a height. In a circle of radius 1 the height of the point at angle θ is sin θ; plotting that height for every angle draws the sine wave.
Formal definition and notation
For an acute angleθ of a right triangle, sin θ is the length of the leg opposite θ divided by the length of the hypotenuse. For any angle θ in standard position, the terminal side meets the unit circlex2 + y2 = 1 at a point (x, y), and sin θ = y. Measuring θ in radians makes sine a function of every real number, y = sinx, with domain all real numbers, range −1 ≤ y ≤ 1 and period2π (that is, 360°).
Symbol
Read it as
Meaning
sin θ
sine theta, or sine of theta
the sine of the angle θ
sin−1x
inverse sine of x, or arcsine of x
the angle whose sine is x; sin−1(0.5) = 30°. It is not 1sinx
sin2θ
sine squared theta
short for (sin θ)2
Key facts
Values to know: sin 0° = 0, sin 30° = 12, sin 45° = √22 ≈ 0.707, sin 60° = √32 ≈ 0.866, sin 90° = 1.
In a right triangle the opposite leg is shorter than the hypotenuse, so the sine of an acute angle is always between 0 and 1.
Sine is positive in Quadrants I and II, where y > 0, and negative in Quadrants III and IV.
Cosine uses the adjacent side: cos θ = adjacenthypotenuse, the x-coordinate on the unit circle. Sine uses the opposite side, the y-coordinate. At 30°, sine is 0.5 and cosine is about 0.866.
Sinevs.Inverse sine (arcsine)
Sine takes an angle and gives a ratio: sin 30° = 0.5. Inverse sine goes the other way, from a ratio to an angle: sin−1(0.5) = 30°.
Tangent compares the two legs, oppositeadjacent, and can be larger than 1. Sine compares a leg with the hypotenuse, so it never goes above 1.
Quick check: 3 questions
Question 1 of 3
What is sin 30°?
Show answer and explanation
Answer: D. In a 30-60-90 triangle the side opposite 30° is half the hypotenuse, so sin 30° = 12. √32 is sin 60°, √22 is sin 45°, and 1 is sin 90°.
Question 2 of 3
Which number cannot be the sine of any angle?
Show answer and explanation
Answer: C. Sine is a y-coordinate on a circle of radius 1, so it is always between −1 and 1. 1.2 is too big. sin 210° = −0.5, sin 0° = 0 and sin 90° = 1.
Question 3 of 3
A right triangle has a 52° angle and a hypotenuse of 10. Which expression gives the side opposite the 52° angle?
Show answer and explanation
Answer: B.sin 52° = x10, so x = 10sin 52° ≈ 7.88. The cosine choice gives the adjacent side, and the two fractions come from solving the proportion the wrong way.
Sine: frequently asked questions
What does sin mean on a calculator?
The sin key gives the sine of the angle you type. In degree mode, sin 30 gives 0.5. In radian mode the same keys find the sine of 30 radians, about −0.988, so check the mode first. The sin−1 key, often reached with Shift or 2nd, does the reverse: it turns a sine value back into an angle.
Why is sine never greater than 1?
In a right triangle the hypotenuse is the longest side, so opposite divided by hypotenuse is less than 1. On the unit circle, sine is the y-coordinate of a point on a circle of radius 1, and no point on that circle is higher than 1 or lower than −1. So −1 ≤ sin θ ≤ 1 for every angle.
What is the sine of 90 degrees?
sin 90° = 1. On the unit circle, the point at 90° is (0, 1), straight up, and its y-coordinate is 1. That is the highest value sine reaches. In a right triangle, as one acute angle grows toward 90°, the opposite leg gets closer and closer to the length of the hypotenuse.
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