In Right-angled Triangle PQR, If P = 60, R = 30 And PR = 12, Then Find The Values Of PQ and QR.

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Introduction

In this article, we will delve into the world of trigonometry and explore how to solve a right-angled triangle using the given information. We will use the concept of trigonometric ratios to find the values of PQ and QR in triangle PQR.

Understanding the Triangle

Given:

  • In right-angled triangle PQR, ∠P = 60°
  • ∠R = 30°
  • PR = 12

Using Trigonometric Ratios

To find the values of PQ and QR, we can use the trigonometric ratios of sine, cosine, and tangent.

  • Sine Ratio: sin(∠P) = PQ / PR
  • Cosine Ratio: cos(∠P) = PR / PQ
  • Tangent Ratio: tan(∠P) = PQ / QR

Finding the Value of PQ

Using the sine ratio, we can find the value of PQ:

sin(60°) = PQ / 12

To find the value of PQ, we can multiply both sides by 12:

PQ = 12 * sin(60°)

Using a calculator, we can find the value of sin(60°):

sin(60°) = 0.866

Now, we can substitute this value into the equation:

PQ = 12 * 0.866 PQ = 10.39

Finding the Value of QR

Using the tangent ratio, we can find the value of QR:

tan(60°) = PQ / QR

We already know the value of PQ (10.39). We can substitute this value into the equation:

tan(60°) = 10.39 / QR

Using a calculator, we can find the value of tan(60°):

tan(60°) = 1.732

Now, we can substitute this value into the equation:

1.732 = 10.39 / QR

To find the value of QR, we can multiply both sides by QR:

QR = 10.39 / 1.732 QR = 6

Conclusion

In this article, we used the concept of trigonometric ratios to find the values of PQ and QR in triangle PQR. We used the sine, cosine, and tangent ratios to solve for the unknown sides of the triangle. By following these steps, we can solve any right-angled triangle using the given information.

Key Takeaways

  • Use the sine, cosine, and tangent ratios to solve for the unknown sides of a right-angled triangle.
  • Use a calculator to find the values of trigonometric ratios.
  • Substitute the values of trigonometric ratios into the equations to solve for the unknown sides.

Real-World Applications

The concept of trigonometry has numerous real-world applications, including:

  • Navigation: Trigonometry is used in navigation to find the distance and direction between two points.
  • Surveying: Trigonometry is used in surveying to find the distance and angle between two points.
  • Physics: Trigonometry is used in physics to describe the motion of objects.

Frequently Asked Questions

  • What is a right-angled triangle? A right-angled triangle is a triangle with one angle equal to 90°.
  • What is the sine ratio? The sine ratio is defined as the ratio of the length of the opposite side to the length of the hypotenuse.
  • What is the tangent ratio? The tangent ratio is defined as the ratio of the length of the opposite side to the length of the adjacent side.

References

  • Trigonometry: A comprehensive guide to trigonometry, including formulas, identities, and applications.
  • Geometry: A comprehensive guide to geometry, including points, lines, angles, and triangles.

Glossary

  • Hypotenuse: The longest side of a right-angled triangle.
  • Opposite side: The side of a right-angled triangle that is opposite to the angle being measured.
  • Adjacent side: The side of a right-angled triangle that is adjacent to the angle being measured.
    Frequently Asked Questions: Trigonometry and Right-Angled Triangles ====================================================================

Q: What is a right-angled triangle?

A: A right-angled triangle is a triangle with one angle equal to 90°. The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs.

Q: What are the three main trigonometric ratios?

A: The three main trigonometric ratios are:

  • Sine ratio: sin(∠P) = PQ / PR
  • Cosine ratio: cos(∠P) = PR / PQ
  • Tangent ratio: tan(∠P) = PQ / QR

Q: How do I use the sine ratio to find the value of PQ?

A: To find the value of PQ using the sine ratio, you can use the following formula:

PQ = PR * sin(∠P)

For example, if PR = 12 and ∠P = 60°, then:

PQ = 12 * sin(60°) PQ = 10.39

Q: How do I use the cosine ratio to find the value of PR?

A: To find the value of PR using the cosine ratio, you can use the following formula:

PR = PQ / cos(∠P)

For example, if PQ = 10.39 and ∠P = 60°, then:

PR = 10.39 / cos(60°) PR = 12

Q: How do I use the tangent ratio to find the value of QR?

A: To find the value of QR using the tangent ratio, you can use the following formula:

QR = PQ / tan(∠P)

For example, if PQ = 10.39 and ∠P = 60°, then:

QR = 10.39 / tan(60°) QR = 6

Q: What is the difference between the sine and cosine ratios?

A: The sine ratio is used to find the value of the opposite side (PQ) when the hypotenuse (PR) is known. The cosine ratio is used to find the value of the hypotenuse (PR) when the opposite side (PQ) is known.

Q: What is the difference between the tangent and cotangent ratios?

A: The tangent ratio is used to find the value of the opposite side (PQ) when the adjacent side (QR) is known. The cotangent ratio is used to find the value of the adjacent side (QR) when the opposite side (PQ) is known.

Q: How do I use trigonometry to solve real-world problems?

A: Trigonometry is used to solve a wide range of real-world problems, including:

  • Navigation: Trigonometry is used in navigation to find the distance and direction between two points.
  • Surveying: Trigonometry is used in surveying to find the distance and angle between two points.
  • Physics: Trigonometry is used in physics to describe the motion of objects.

Q: What are some common mistakes to avoid when using trigonometry?

A: Some common mistakes to avoid when using trigonometry include:

  • Using the wrong trigonometric ratio: Make sure to use the correct trigonometric ratio for the problem you are trying to solve.
  • Not checking units: Make sure to check the units of the values you are using to ensure that they are consistent.
  • Not using a calculator: Make sure to use a calculator to find the values of trigonometric ratios.

Q: How do I choose the right trigonometric ratio for a problem?

A: To choose the right trigonometric ratio for a problem, you need to identify the type of problem you are trying to solve. For example:

  • If you are trying to find the value of the opposite side (PQ) when the hypotenuse (PR) is known, use the sine ratio.
  • If you are trying to find the value of the hypotenuse (PR) when the opposite side (PQ) is known, use the cosine ratio.
  • If you are trying to find the value of the adjacent side (QR) when the opposite side (PQ) is known, use the tangent ratio.

Q: What are some real-world applications of trigonometry?

A: Trigonometry has numerous real-world applications, including:

  • Navigation: Trigonometry is used in navigation to find the distance and direction between two points.
  • Surveying: Trigonometry is used in surveying to find the distance and angle between two points.
  • Physics: Trigonometry is used in physics to describe the motion of objects.

Q: How do I use trigonometry to solve problems in physics?

A: Trigonometry is used in physics to describe the motion of objects. To use trigonometry to solve problems in physics, you need to identify the type of problem you are trying to solve. For example:

  • If you are trying to find the distance traveled by an object, use the sine and cosine ratios.
  • If you are trying to find the angle of a projectile, use the tangent ratio.

Q: What are some common mistakes to avoid when using trigonometry in physics?

A: Some common mistakes to avoid when using trigonometry in physics include:

  • Using the wrong trigonometric ratio: Make sure to use the correct trigonometric ratio for the problem you are trying to solve.
  • Not checking units: Make sure to check the units of the values you are using to ensure that they are consistent.
  • Not using a calculator: Make sure to use a calculator to find the values of trigonometric ratios.