# SINE AND ARC SINE A TEMPORARY MATHEMATICAL DETOUR.

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SINE AND ARC SINE A TEMPORARY MATHEMATICAL DETOUR

SIDES OF A RIGHT ANGLE TRIANGLES A right angle triangle has 3 sides, and 2 sides are perpendicular to each other The longest side of a right angle triangle is called the hypotenuse. Hypotenuse

SIDES OF A RIGHT ANGLE TRIANGLE Other than the hypotenuse, the other sides of a right angle triangle must be made in reference to an angle: Consider the triangle below: x Hypotenuse Opposite Adjacent y Hypotenuse Opposite Adjacent

SINE There are 3 trigonometric functions: sine, cosine and tangent. In Physics, you will only need to use sine. (short form: sin) The Sine of an angle is defined as length of opposite side divided by length of hypotenuse x Hypotenuse Opposite Adjacent

SINE For example, consider triangle below What is sin x? x 5 m 4 m Opposite side = 4 m Hypotheneus = 5 m Recall: Sin x = opposite / hypotheneus = 4/5 = 0.8 (no units!) Note: You can only Sine an angle Trigo identities have no units Sine and Cosine are always less than 1

ARC SINE Arc Sine is the opposite of Sine (sometimes called “inverse sine”) Symbol of Arc Sine is sin -1 For Sine, I give you an angle, and you give me a ratio (i.e. length of opposite side divided by hypotheneus) For Arc Sine, I give you the ratio, you give me the angle x 5 m 4 m Sin x = 0.8 Sin -1 (0.8) = x (in degrees) [Can you find Sin-1 using your calculator?] Sin -1 (0.8) = 53.1° (1 decimal place) Note: this presentation is okay for Physics exam, but is not allowed for Math exam!!

TEST YOURSELF Find Angle x using Arc Sine x 10 m 6 m Sin x = 6/10 = 0.6 Sin -1 (0.6) = x [Use calculator] X = 36.9° (1 decimal place)