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Sagot :
Given:
[tex]\sin (u)=-\dfrac{7}{25}[/tex]
[tex]\cos (v)=-\dfrac{4}{5}[/tex]
To find:
The exact value of cos(u-v) if both angles are in quadrant 3.
Solution:
In 3rd quadrant, cos and sin both trigonometric ratios are negative.
We have,
[tex]\sin (u)=-\dfrac{7}{25}[/tex]
[tex]\cos (v)=-\dfrac{4}{5}[/tex]
Now,
[tex]\cos (u)=-\sqrt{1-\sin^2 (u)}[/tex]
[tex]\cos (u)=-\sqrt{1-(-\dfrac{7}{25})^2}[/tex]
[tex]\cos (u)=-\sqrt{1-\dfrac{49}{625}}[/tex]
[tex]\cos (u)=-\sqrt{\dfrac{625-49}{625}}[/tex]
On further simplification, we get
[tex]\cos (u)=-\sqrt{\dfrac{576}{625}}[/tex]
[tex]\cos (u)=-\dfrac{24}{25}[/tex]
Similarly,
[tex]\sin (v)=-\sqrt{1-\cos^2 (v)}[/tex]
[tex]\sin (v)=-\sqrt{1-(-\dfrac{4}{5})^2}[/tex]
[tex]\sin (v)=-\sqrt{1-\dfrac{16}{25}}[/tex]
[tex]\sin (v)=-\sqrt{\dfrac{25-16}{25}}[/tex]
[tex]\sin (v)=-\sqrt{\dfrac{9}{25}}[/tex]
[tex]\sin (v)=-\dfrac{3}{5}[/tex]
Now,
[tex]\cos (u-v)=\cos u\cos v+\sin u\sin v[/tex]
[tex]\cos (u-v)=\left(-\dfrac{24}{25}\right)\left(-\dfrac{4}{5}\right)+\left(-\dfrac{7}{25}\right)\left(-\dfrac{3}{25}\right)[/tex]
[tex]\cos (u-v)=\dfrac{96}{625}+\dfrac{21}{625}[/tex]
[tex]\cos (u-v)=\dfrac{1 17}{625}[/tex]
Therefore, the value of cos (u-v) is 0.1872.
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