Chapter 13.7, Problem 2CE

To determine

Calculate the value of the angle using sine, cosine and tangent addition formula.

Answer to Problem 2CE

The solution is,

  

Explanation of Solution

Given:The sine addition formula is,

  $\sin 10°=\sin (5°+5°)=\sin 5°\cos 5°+\cos 5°\sin 5°$

And the value of $\sin 5°$ is,

  $\sin 5°=0.087155743$

Calculation:

Let’s take the angles,

  $angle=5°,10°,15°,20°,25°,30°,35°,40°,45°,50°,55°,60°,65°,70°,75°,80°,85°,90°$

Then use addition rule of sine, cosine and tangent,

  $\begin{array}{l}\sin (a+b)=\sin (a+b)=\sin a\cos b+\cos a\sin b \mathrm{......}(1)\\ \cos (a+b)=\cos a\cos b-\sin a\sin b \mathrm{......}(2)\\ \tan (a+b)=\frac{\tan a+\tan b}{1-\tan a\tan b} \mathrm{......}(3)\end{array}$

Calculate the $\sin 10°$ ,

  $\sin 10°=\sin (5°+5°)=\sin 5°\cos 5°+\cos 5°\sin 5° \mathrm{......}(4)$

Find $\cos 5°$ ,

  $\cos 5°=\sqrt{1-{\sin }^{2}5°}$

Put the value of $\cos 5°$ into equation (4) and get,

  $\sin 10°=\text{0}\text{.17365}$

Similarly, repeat for cosine and tangent,

  $\begin{array}{l}\cos (5°+5°)=\cos 5°\cos 5°-\sin 5°\sin 5°=\text{0}\text{.98481}\\ \tan (5°+5°)=\frac{\tan 5°+\tan 5°}{1-\tan 5°\tan 5°}=\text{0}\text{.17565}\end{array}$

program:

function [S,C,t] = sinadd(a,b)
% funtion for sin(a+b)
cosd(a)=sqrt(1-sind(a).^2);
cosd(b)=sqrt(1-sind(b).^2);
S=sind(a)*cosd(b)+cosd(a)*sind(b);
C=cosd(a)*cosd(b)-sind(a)*sind(b);
t=tand(a)+tand(b)./(1-tand(a)*tand(b));
end

clc
clear
close all
a=5; b=(10:5:90)-a;
for i=1:17
    [Sin(i),Cos(i),t(i)]=sinadd(a,b(i));
end
Sin=Sin';
Cos=Cos';
Tan=t';
angle=(a+b)';
T1= table(angle,Sin,Cos,Tan)

Query:

• First, define the angles.
• Then calculate value of each angle using function “sinadd”.
• Then convert into tabulate form.