# 1. If know x = 112^{o}: find cosx ; sinx? 2. Find sin1^{o}.

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### 3 Answers

There are two taylor series you can use to solve sine and cosine without a calculator or tables:

**Sine**

x - (x^3/3!) + (x^5/5!) - ...

**Cosine**

1 - (x^2/2!) + (x^4/4!) - ...

Just sub your degree (in radians) into x, and solve to however many degrees of accuracy you want.

x= 112 degrees.

==> cos 112 = -0.3746

.==> sin 112 = sqrt(1-cos^2 112)

= sqrt(1-0.1403)

= sqrt(0.8597)

==> sin 112= 0.9272

2) sin1 = 0.0175

1)

If x= 112 degree = 90+22 degree= 180-68 degree.

sin(90+22) = sin (180 - 68) = sin68 = 0.9272 using tables.

For approximation, sinx = 4x(180-x)/(40500-x(180-x), for 0<x<180 only.

= 4*112*68/(40500-112*68) = 0.9264, cosx = sqrt(1-sin^2 x) = -0.3765

cos112 = cos(180-68) = -cos68 = -0.3746 using tables

2)

sin 1 degree = sin (pi/180 radians) = sin (0.1745 radians) = 0.017452, as for small x in radians, sinx --> x.