If sin A = 3/4, calculate for cos A. Choose the correct matching and determine which of the statements is TRUE: I: The expression of tan A in terms of sin A is sin A / √(1 + sin² A... If sin A = 3/4, calculate for cos A. Choose the correct matching and determine which of the statements is TRUE: I: The expression of tan A in terms of sin A is sin A / √(1 + sin² A). II: The value of sin A or cos A never exceeds one.
Understand the Problem
The question is asking to solve problems related to trigonometry, including calculating the cosine of an angle given its sine value, matching trigonometric identities, and determining the truth of statements related to trigonometric ratios.
Answer
The value of \( \cos A \) is \( \frac{\sqrt{7}}{4} \) and both statements are true.
Answer for screen readers
$$ \cos A = \frac{\sqrt{7}}{4} $$
Both statements I and II are true.
Steps to Solve
- Calculate cos A using the Pythagorean identity
Given that ( \sin A = \frac{3}{4} ), use the Pythagorean identity: $$ \sin^2 A + \cos^2 A = 1 $$
Plugging in the value of ( \sin A ): $$ \left(\frac{3}{4}\right)^2 + \cos^2 A = 1 $$
So, $$ \frac{9}{16} + \cos^2 A = 1 $$
- Rearrange the equation to solve for cos A
Subtract ( \frac{9}{16} ) from both sides: $$ \cos^2 A = 1 - \frac{9}{16} $$
Convert 1 to a fraction: $$ \cos^2 A = \frac{16}{16} - \frac{9}{16} = \frac{7}{16} $$
- Take the square root
Taking the square root gives: $$ \cos A = \sqrt{\frac{7}{16}} $$
Which simplifies to: $$ \cos A = \frac{\sqrt{7}}{4} $$
- Identify true statements
For the statements:
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I: The expression of ( \tan A ) in terms of ( \sin A ): $$ \tan A = \frac{\sin A}{\sqrt{1 - \sin^2 A}} $$ This expression is correct.
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II: The value of ( \sin A ) or ( \cos A ) never exceeds one. This statement is also true since both sine and cosine values range from -1 to 1.
Thus, both statements I and II are true.
$$ \cos A = \frac{\sqrt{7}}{4} $$
Both statements I and II are true.
More Information
In this problem, we calculate ( \cos A ) using a fundamental trigonometric identity. The sine and cosine functions are crucial in various applications of trigonometry, and they help relate angles to the sides of right triangles.
Tips
- Confusing sine and cosine values: Always remember that sine and cosine are related through the Pythagorean identity.
- Taking square roots: When calculating ( \cos A ), students might forget to consider both the positive and negative roots, but in the context of angles, use the appropriate quadrant.
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