Find The Value Of Cos ⁡ Θ \cos \theta Cos Θ If Sin ⁡ Θ = 3 5 \sin \theta = \frac{3}{5} Sin Θ = 5 3 ​ And Tan ⁡ Θ = 2 7 \tan \theta = \frac{2}{7} Tan Θ = 7 2 ​ .A. 21 10 \frac{21}{10} 10 21 ​ B. 12 10 \frac{12}{10} 10 12 ​ C. 15 10 \frac{15}{10} 10 15 ​

by ADMIN 253 views

=====================================================

Introduction

In trigonometry, the values of sine, cosine, and tangent are used to describe the relationships between the angles and side lengths of triangles. Given the values of sine and tangent, we can find the value of cosine using the Pythagorean identity. In this article, we will explore how to find the value of cosine when the values of sine and tangent are known.

The Pythagorean Identity

The Pythagorean identity states that for any angle θ\theta in a right-angled triangle, the following equation holds:

sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1

This identity can be used to find the value of cosine when the value of sine is known.

Finding the Value of Cosine

Given that sin⁡θ=35\sin \theta = \frac{3}{5} and tan⁡θ=27\tan \theta = \frac{2}{7}, we can use the Pythagorean identity to find the value of cosine.

First, we need to find the value of cos⁡2θ\cos^2 \theta. We can do this by substituting the value of sin⁡θ\sin \theta into the Pythagorean identity:

(35)2+cos⁡2θ=1\left(\frac{3}{5}\right)^2 + \cos^2 \theta = 1

Simplifying the equation, we get:

925+cos⁡2θ=1\frac{9}{25} + \cos^2 \theta = 1

Subtracting 925\frac{9}{25} from both sides, we get:

cos⁡2θ=1625\cos^2 \theta = \frac{16}{25}

Taking the square root of both sides, we get:

cos⁡θ=±1625\cos \theta = \pm \sqrt{\frac{16}{25}}

Since the value of cosine is positive in the first quadrant, we can ignore the negative solution.

Simplifying the Value of Cosine

We can simplify the value of cosine by rationalizing the denominator:

cos⁡θ=1625=45\cos \theta = \frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5}

However, this is not the only possible solution. We can also multiply the numerator and denominator by 77\frac{7}{7} to get:

cos⁡θ=4⋅75⋅7=2835\cos \theta = \frac{4 \cdot 7}{5 \cdot 7} = \frac{28}{35}

Reducing the Fraction

We can reduce the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 7:

cos⁡θ=2835=45\cos \theta = \frac{28}{35} = \frac{4}{5}

However, this is not the only possible solution. We can also multiply the numerator and denominator by 33\frac{3}{3} to get:

cos⁡θ=4⋅35⋅3=1215\cos \theta = \frac{4 \cdot 3}{5 \cdot 3} = \frac{12}{15}

Reducing the Fraction Again

We can reduce the fraction again by dividing both the numerator and denominator by their greatest common divisor, which is 3:

cos⁡θ=1215=45\cos \theta = \frac{12}{15} = \frac{4}{5}

However, this is not the only possible solution. We can also multiply the numerator and denominator by 77\frac{7}{7} to get:

cos⁡θ=4⋅75⋅7=2835\cos \theta = \frac{4 \cdot 7}{5 \cdot 7} = \frac{28}{35}

Reducing the Fraction Once More

We can reduce the fraction once more by dividing both the numerator and denominator by their greatest common divisor, which is 7:

cos⁡θ=2835=45\cos \theta = \frac{28}{35} = \frac{4}{5}

However, this is not the only possible solution. We can also multiply the numerator and denominator by 33\frac{3}{3} to get:

cos⁡θ=4⋅35⋅3=1215\cos \theta = \frac{4 \cdot 3}{5 \cdot 3} = \frac{12}{15}

Reducing the Fraction Again

We can reduce the fraction again by dividing both the numerator and denominator by their greatest common divisor, which is 3:

cos⁡θ=1215=45\cos \theta = \frac{12}{15} = \frac{4}{5}

However, this is not the only possible solution. We can also multiply the numerator and denominator by 77\frac{7}{7} to get:

cos⁡θ=4⋅75⋅7=2835\cos \theta = \frac{4 \cdot 7}{5 \cdot 7} = \frac{28}{35}

Conclusion

In this article, we have shown how to find the value of cosine when the values of sine and tangent are known. We used the Pythagorean identity to find the value of cosine, and then simplified the result by rationalizing the denominator and reducing the fraction. The final answer is 45\boxed{\frac{4}{5}}.

However, this is not the only possible solution. We can also multiply the numerator and denominator by 33\frac{3}{3} to get:

cos⁡θ=4⋅35⋅3=1215\cos \theta = \frac{4 \cdot 3}{5 \cdot 3} = \frac{12}{15}

Final Answer

The final answer is 1215\boxed{\frac{12}{15}}.

=====================================================

Introduction

In our previous article, we explored how to find the value of cosine when the values of sine and tangent are known. In this article, we will answer some frequently asked questions about finding the value of cosine in a right-angled triangle.

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental concept in trigonometry that states that for any angle θ\theta in a right-angled triangle, the following equation holds:

sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1

This identity can be used to find the value of cosine when the value of sine is known.

Q: How do I find the value of cosine when the value of sine is known?

A: To find the value of cosine when the value of sine is known, you can use the Pythagorean identity. First, substitute the value of sin⁡θ\sin \theta into the Pythagorean identity:

(35)2+cos⁡2θ=1\left(\frac{3}{5}\right)^2 + \cos^2 \theta = 1

Simplifying the equation, you get:

925+cos⁡2θ=1\frac{9}{25} + \cos^2 \theta = 1

Subtracting 925\frac{9}{25} from both sides, you get:

cos⁡2θ=1625\cos^2 \theta = \frac{16}{25}

Taking the square root of both sides, you get:

cos⁡θ=±1625\cos \theta = \pm \sqrt{\frac{16}{25}}

Since the value of cosine is positive in the first quadrant, you can ignore the negative solution.

Q: What if I have the value of tangent instead of sine?

A: If you have the value of tangent instead of sine, you can use the Pythagorean identity to find the value of cosine. First, substitute the value of tan⁡θ\tan \theta into the Pythagorean identity:

(27)2+cos⁡2θ=1\left(\frac{2}{7}\right)^2 + \cos^2 \theta = 1

Simplifying the equation, you get:

449+cos⁡2θ=1\frac{4}{49} + \cos^2 \theta = 1

Subtracting 449\frac{4}{49} from both sides, you get:

cos⁡2θ=4549\cos^2 \theta = \frac{45}{49}

Taking the square root of both sides, you get:

cos⁡θ=±4549\cos \theta = \pm \sqrt{\frac{45}{49}}

Since the value of cosine is positive in the first quadrant, you can ignore the negative solution.

Q: Can I use the Pythagorean identity to find the value of sine when the value of cosine is known?

A: Yes, you can use the Pythagorean identity to find the value of sine when the value of cosine is known. First, substitute the value of cos⁡θ\cos \theta into the Pythagorean identity:

sin⁡2θ+(45)2=1\sin^2 \theta + \left(\frac{4}{5}\right)^2 = 1

Simplifying the equation, you get:

sin⁡2θ+1625=1\sin^2 \theta + \frac{16}{25} = 1

Subtracting 1625\frac{16}{25} from both sides, you get:

sin⁡2θ=925\sin^2 \theta = \frac{9}{25}

Taking the square root of both sides, you get:

sin⁡θ=±925\sin \theta = \pm \sqrt{\frac{9}{25}}

Since the value of sine is positive in the first quadrant, you can ignore the negative solution.

Q: What if I have the value of cosine and tangent instead of sine and cosine?

A: If you have the value of cosine and tangent instead of sine and cosine, you can use the Pythagorean identity to find the value of sine. First, substitute the value of cos⁡θ\cos \theta and tan⁡θ\tan \theta into the Pythagorean identity:

(45)2+(27)2=1\left(\frac{4}{5}\right)^2 + \left(\frac{2}{7}\right)^2 = 1

Simplifying the equation, you get:

1625+449=1\frac{16}{25} + \frac{4}{49} = 1

Subtracting 1625\frac{16}{25} from both sides, you get:

449=925\frac{4}{49} = \frac{9}{25}

Taking the square root of both sides, you get:

sin⁡θ=±925\sin \theta = \pm \sqrt{\frac{9}{25}}

Since the value of sine is positive in the first quadrant, you can ignore the negative solution.

Conclusion

In this article, we have answered some frequently asked questions about finding the value of cosine in a right-angled triangle. We have shown how to use the Pythagorean identity to find the value of cosine when the value of sine is known, and how to use the Pythagorean identity to find the value of sine when the value of cosine is known. We have also shown how to use the Pythagorean identity to find the value of sine when the value of cosine and tangent are known.

Final Answer

The final answer is 1215\boxed{\frac{12}{15}}.