Problem 27 - Contemporary Precalculus

Question

Use the Pythagorean identity to find sin \(t\). $$\cos t=1 / 2 \quad \text { and } \quad 0

Step-by-Step Solution

Answer

Answer: \(\sin t=\frac{\sqrt{3}}{2}\)

Step 1: Write the Pythagorean Identity

Recall the Pythagorean identity: $$\sin^2 t + \cos^2 t = 1$$

Step 2: Substitute the given value of \(\cos t\)

We are given that \(\cos t = \frac{1}{2}\), so we substitute this into the identity: $$\sin^2 t + \left(\frac{1}{2}\right)^2 = 1$$

Step 3: Simplify the Identity

Now we simplify the equation: $$\sin^2 t + \frac{1}{4} = 1$$

Step 4: Solve for \(\sin^2 t\)

To solve for \(\sin^2 t\), we'll subtract \(\frac{1}{4}\) from both sides of the equation: $$\sin^2 t = 1 - \frac{1}{4}$$

Step 5: Simplify the Expression

Simplify the expression on the right side of the equation: $$\sin^2 t = \frac{3}{4}$$

Step 6: Solve for \(\sin t\)

To find the value of \(\sin t\), take the square root of both sides of the equation. Since \(t\) is in the first quadrant (0

Step 7: Simplify the Final Expression

Simplify the square root: $$\sin t = \frac{\sqrt{3}}{2}$$ So, \(\sin t=\frac{\sqrt{3}}{2}\).

Key Concepts

Pythagorean Identity

The Pythagorean identity is one of the fundamental tools in trigonometry. It's very similar to the Pythagorean theorem, which relates the lengths of the sides of a right triangle. In the world of trigonometry, this identity shows the relationship between the square of the sine and cosine of an angle. The standard form of this identity is:\[\sin^2 t + \cos^2 t = 1\]This equation tells us that no matter what angle you have, the square of the sine plus the square of the cosine will always equal 1. This property is incredibly helpful for solving problems, allowing us to find one trigonometric function if we know the other. The identity holds true for all angles, not just those in the first quadrant.

Sine Function

The sine function is a fundamental trigonometric function that describes the ratio of the length of the opposite side of a right triangle to its hypotenuse.

• It is represented as \( \sin \theta \), where \( \theta \) is the angle.
• The sine of an angle in the first quadrant is always positive.
• It ranges from 0 to 1 as the angle moves from 0 to \( \frac{\pi}{2} \).

In the context of this problem, knowing that the cosine of the angle is \( \frac{1}{2} \) in the first quadrant helps us use the Pythagorean identity to find the sine. By substituting \( \cos t \) with \( \frac{1}{2} \), we calculated \( \sin t \) to be \( \frac{\sqrt{3}}{2} \). This is consistent with the positive nature of the sine function in the first quadrant.

Cosine Function

The cosine function is another essential trigonometric function, describing the ratio of the length of the adjacent side of a right triangle to its hypotenuse.

• Represented as \( \cos \theta \), with \( \theta \) as the angle.
• The cosine of an angle in the first quadrant is always positive.
• It ranges from 1 to 0 as the angle increases from 0 to \( \frac{\pi}{2} \).

In this exercise, knowing that \( \cos t = \frac{1}{2} \) helps us find the sine using the Pythagorean identity. Cosine provides a way to transition between the xy-coordinates of a unit circle, and plays a key role in finding other trigonometric functions through identities.

First Quadrant

The first quadrant of the coordinate system is where both x and y coordinates are positive. This concept is vital in understanding the signs of trigonometric functions.

• The angles here range from 0 to \( \frac{\pi}{2} \).
• Both sine and cosine functions are positive in this quadrant.
• This quadrant represents the top-right section of a coordinate plane.

In the problem context, knowing that \( t \) is between 0 and \( \frac{\pi}{2} \) confirms that both \( \sin t \) and \( \cos t \) are positive. This makes problem-solving straightforward when using identities like the Pythagorean identity, and provides confidence in the calculated values of these functions.

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Problem 27

Other exercises in this chapter

Problem 26

In Exercises \(15-29,\) find the exact value of the sine, cosine, and tangent of the number, without using a calculator. $$-10 \pi / 3$$

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Problem 27

Sketch a complete graph of the function. $$k(t)=-3 \sin t$$

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Problem 27

Convert the given degree measure to radians. $$6^{\circ}$$

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Problem 27

In Exercises \(15-29,\) find the exact value of the sine, cosine, and tangent of the number, without using a calculator. $$-15 \pi / 4$$

View solution

Key Concepts

Pythagorean Identity

The Pythagorean identity is one of the fundamental tools in trigonometry. It's very similar to the Pythagorean theorem, which relates the lengths of the sides of a right triangle. In the world of trigonometry, this identity shows the relationship between the square of the sine and cosine of an angle. The standard form of this identity is:\[\sin^2 t + \cos^2 t = 1\]This equation tells us that no matter what angle you have, the square of the sine plus the square of the cosine will always equal 1. This property is incredibly helpful for solving problems, allowing us to find one trigonometric function if we know the other. The identity holds true for all angles, not just those in the first quadrant.

Sine Function

The sine function is a fundamental trigonometric function that describes the ratio of the length of the opposite side of a right triangle to its hypotenuse.

• It is represented as \( \sin \theta \), where \( \theta \) is the angle.
• The sine of an angle in the first quadrant is always positive.
• It ranges from 0 to 1 as the angle moves from 0 to \( \frac{\pi}{2} \).

In the context of this problem, knowing that the cosine of the angle is \( \frac{1}{2} \) in the first quadrant helps us use the Pythagorean identity to find the sine. By substituting \( \cos t \) with \( \frac{1}{2} \), we calculated \( \sin t \) to be \( \frac{\sqrt{3}}{2} \). This is consistent with the positive nature of the sine function in the first quadrant.

Cosine Function

The cosine function is another essential trigonometric function, describing the ratio of the length of the adjacent side of a right triangle to its hypotenuse.

• Represented as \( \cos \theta \), with \( \theta \) as the angle.
• The cosine of an angle in the first quadrant is always positive.
• It ranges from 1 to 0 as the angle increases from 0 to \( \frac{\pi}{2} \).

In this exercise, knowing that \( \cos t = \frac{1}{2} \) helps us find the sine using the Pythagorean identity. Cosine provides a way to transition between the xy-coordinates of a unit circle, and plays a key role in finding other trigonometric functions through identities.

First Quadrant

The first quadrant of the coordinate system is where both x and y coordinates are positive. This concept is vital in understanding the signs of trigonometric functions.

• The angles here range from 0 to \( \frac{\pi}{2} \).
• Both sine and cosine functions are positive in this quadrant.
• This quadrant represents the top-right section of a coordinate plane.

In the problem context, knowing that \( t \) is between 0 and \( \frac{\pi}{2} \) confirms that both \( \sin t \) and \( \cos t \) are positive. This makes problem-solving straightforward when using identities like the Pythagorean identity, and provides confidence in the calculated values of these functions.