When To Use Tan Sin Or Cos

Let's delve into the world of trigonometry and clarify when to use tan, sin, or cos functions. These three trigonometric functions, often abbreviated as tan, sin, and cos, are fundamental tools for analyzing triangles and solving problems involving angles and side lengths. Understanding when to apply each function is crucial for success in trigonometry and related fields.

Introduction to Trigonometric Ratios

Sine (sin), cosine (cos), and tangent (tan) are trigonometric ratios that relate the angles of a right triangle to the lengths of its sides. They are defined based on the relationships between the opposite side, adjacent side, and hypotenuse of a right triangle relative to a specific acute angle (an angle less than 90 degrees).

Before diving into when to use each function, let's define the terms:

• Hypotenuse: The longest side of a right triangle, opposite the right angle (90 degrees).
• Opposite Side: The side opposite to the angle of interest (the angle we are working with).
• Adjacent Side: The side adjacent to the angle of interest (the angle we are working with), which is not the hypotenuse.

With these definitions in mind, we can define the three trigonometric ratios:

• Sine (sin): The ratio of the length of the opposite side to the length of the hypotenuse. sin(θ) = Opposite / Hypotenuse
• Cosine (cos): The ratio of the length of the adjacent side to the length of the hypotenuse. cos(θ) = Adjacent / Hypotenuse
• Tangent (tan): The ratio of the length of the opposite side to the length of the adjacent side. tan(θ) = Opposite / Adjacent

A helpful mnemonic to remember these relationships is SOH CAH TOA:

• SOH: Sine = Opposite / Hypotenuse
• CAH: Cosine = Adjacent / Hypotenuse
• TOA: Tangent = Opposite / Adjacent

When to Use Sine (sin)

The sine function is used when you have information about the opposite side and the hypotenuse of a right triangle, or when you need to find one of these values. Here's a breakdown of the scenarios:

• Given the angle and the hypotenuse, find the opposite side: If you know the angle (θ) and the length of the hypotenuse, you can use the sine function to find the length of the opposite side.

  • sin(θ) = Opposite / Hypotenuse
  • Opposite = sin(θ) * Hypotenuse

• Given the angle and the opposite side, find the hypotenuse: If you know the angle (θ) and the length of the opposite side, you can use the sine function to find the length of the hypotenuse.

  • sin(θ) = Opposite / Hypotenuse
  • Hypotenuse = Opposite / sin(θ)

• Given the opposite side and the hypotenuse, find the angle: If you know the lengths of the opposite side and the hypotenuse, you can use the inverse sine function (arcsin or sin⁻¹) to find the angle (θ).

  • sin(θ) = Opposite / Hypotenuse
  • θ = arcsin(Opposite / Hypotenuse)

Examples:

1. Finding the opposite side: A ladder leans against a wall at an angle of 60 degrees with the ground. The ladder is 10 feet long. How high up the wall does the ladder reach?

   • We know the angle (θ = 60°) and the hypotenuse (ladder length = 10 ft). We want to find the opposite side (height on the wall).
   • sin(60°) = Opposite / 10
   • Opposite = sin(60°) * 10
   • Opposite ≈ 0.866 * 10
   • Opposite ≈ 8.66 feet
   • The ladder reaches approximately 8.66 feet up the wall.

2. Finding the hypotenuse: A kite is flying at an angle of 45 degrees with the ground. The kite is directly above a point on the ground 30 feet away from the person holding the string. What is the length of the kite string, assuming it is straight? (This actually requires tangent, but we will adjust the scenario) Instead, let's say the kite is directly vertically 30 feet above the person holding the string. What is the length of the kite string, assuming it is straight?

   • We know the angle (θ = 45°) and the opposite side (height of the kite = 30 ft). We want to find the hypotenuse (length of the string).
   • sin(45°) = 30 / Hypotenuse
   • Hypotenuse = 30 / sin(45°)
   • Hypotenuse ≈ 30 / 0.707
   • Hypotenuse ≈ 42.43 feet
   • The length of the kite string is approximately 42.43 feet.

3. Finding the angle: A ramp rises 3 feet over a horizontal distance of 5 feet to reach a door. What is the angle of elevation of the ramp? (Actually requires tangent, but we will adjust.) Instead, the ramp has a length of 5 feet and rises 3 feet.

   • We know the opposite side (height of the ramp = 3 ft) and the hypotenuse (length of the ramp = 5 ft). We want to find the angle (θ).
   • sin(θ) = 3 / 5
   • θ = arcsin(3 / 5)
   • θ ≈ arcsin(0.6)
   • θ ≈ 36.87 degrees
   • The angle of elevation of the ramp is approximately 36.87 degrees.

When to Use Cosine (cos)

The cosine function is used when you have information about the adjacent side and the hypotenuse of a right triangle, or when you need to find one of these values. Here's a breakdown of the scenarios:

• Given the angle and the hypotenuse, find the adjacent side: If you know the angle (θ) and the length of the hypotenuse, you can use the cosine function to find the length of the adjacent side.

  • cos(θ) = Adjacent / Hypotenuse
  • Adjacent = cos(θ) * Hypotenuse

• Given the angle and the adjacent side, find the hypotenuse: If you know the angle (θ) and the length of the adjacent side, you can use the cosine function to find the length of the hypotenuse.

  • cos(θ) = Adjacent / Hypotenuse
  • Hypotenuse = Adjacent / cos(θ)

• Given the adjacent side and the hypotenuse, find the angle: If you know the lengths of the adjacent side and the hypotenuse, you can use the inverse cosine function (arccos or cos⁻¹) to find the angle (θ).

  • cos(θ) = Adjacent / Hypotenuse
  • θ = arccos(Adjacent / Hypotenuse)

Examples:

1. Finding the adjacent side: A 15-foot ladder leans against a wall at an angle of 70 degrees with the ground. How far is the base of the ladder from the wall?

   • We know the angle (θ = 70°) and the hypotenuse (ladder length = 15 ft). We want to find the adjacent side (distance from the wall).
   • cos(70°) = Adjacent / 15
   • Adjacent = cos(70°) * 15
   • Adjacent ≈ 0.342 * 15
   • Adjacent ≈ 5.13 feet
   • The base of the ladder is approximately 5.13 feet from the wall.

2. Finding the hypotenuse: A cable is anchored to the ground 10 feet away from the base of a pole. The cable makes an angle of 65 degrees with the ground. How long is the cable?

   • We know the angle (θ = 65°) and the adjacent side (distance from the pole = 10 ft). We want to find the hypotenuse (length of the cable).
   • cos(65°) = 10 / Hypotenuse
   • Hypotenuse = 10 / cos(65°)
   • Hypotenuse ≈ 10 / 0.423
   • Hypotenuse ≈ 23.64 feet
   • The length of the cable is approximately 23.64 feet.

3. Finding the angle: A slide in a park is 8 feet long and its base extends 6 feet from the base of the platform. What is the angle the slide makes with the ground?

   • We know the adjacent side (base length = 6 ft) and the hypotenuse (slide length = 8 ft). We want to find the angle (θ).
   • cos(θ) = 6 / 8
   • θ = arccos(6 / 8)
   • θ = arccos(0.75)
   • θ ≈ 41.41 degrees
   • The angle the slide makes with the ground is approximately 41.41 degrees.

When to Use Tangent (tan)

The tangent function is used when you have information about the opposite side and the adjacent side of a right triangle, or when you need to find one of these values. Here's a breakdown of the scenarios:

• Given the angle and the adjacent side, find the opposite side: If you know the angle (θ) and the length of the adjacent side, you can use the tangent function to find the length of the opposite side.

  • tan(θ) = Opposite / Adjacent
  • Opposite = tan(θ) * Adjacent

• Given the angle and the opposite side, find the adjacent side: If you know the angle (θ) and the length of the opposite side, you can use the tangent function to find the length of the adjacent side.

  • tan(θ) = Opposite / Adjacent
  • Adjacent = Opposite / tan(θ)

• Given the opposite side and the adjacent side, find the angle: If you know the lengths of the opposite side and the adjacent side, you can use the inverse tangent function (arctan or tan⁻¹) to find the angle (θ).

  • tan(θ) = Opposite / Adjacent
  • θ = arctan(Opposite / Adjacent)

Examples:

1. Finding the opposite side: A building casts a shadow 40 feet long. The angle of elevation of the sun is 50 degrees. How tall is the building?

   • We know the angle (θ = 50°) and the adjacent side (shadow length = 40 ft). We want to find the opposite side (building height).
   • tan(50°) = Opposite / 40
   • Opposite = tan(50°) * 40
   • Opposite ≈ 1.192 * 40
   • Opposite ≈ 47.68 feet
   • The building is approximately 47.68 feet tall.

2. Finding the adjacent side: A surveyor stands on top of a cliff 100 feet above sea level. He spots a boat at an angle of depression of 25 degrees. How far is the boat from the base of the cliff?

   • We know the angle (θ = 25°) and the opposite side (cliff height = 100 ft). We want to find the adjacent side (distance from the base). Note: the angle of depression is the angle from the horizontal down to the boat, so the angle inside the triangle at the boat is also 25 degrees (alternate interior angles).
   • tan(25°) = 100 / Adjacent
   • Adjacent = 100 / tan(25°)
   • Adjacent ≈ 100 / 0.466
   • Adjacent ≈ 214.59 feet
   • The boat is approximately 214.59 feet from the base of the cliff.

3. Finding the angle: A ladder leans against a wall. The base of the ladder is 5 feet from the wall, and the ladder reaches a height of 12 feet on the wall. What angle does the ladder make with the ground?

   • We know the opposite side (height on the wall = 12 ft) and the adjacent side (distance from the wall = 5 ft). We want to find the angle (θ).
   • tan(θ) = 12 / 5
   • θ = arctan(12 / 5)
   • θ = arctan(2.4)
   • θ ≈ 67.38 degrees
   • The ladder makes an angle of approximately 67.38 degrees with the ground.

Choosing the Right Function: A Summary

To summarize, here's a quick guide to choosing the right trigonometric function:

• Sine (sin): Use when you have information about the opposite side and the hypotenuse.
• Cosine (cos): Use when you have information about the adjacent side and the hypotenuse.
• Tangent (tan): Use when you have information about the opposite side and the adjacent side.

When solving problems, always:

1. Draw a diagram: Sketch the right triangle and label the known and unknown quantities.
2. Identify the angle of interest: Determine which angle you're working with.
3. Determine the known sides: Identify which sides are given relative to the angle (opposite, adjacent, hypotenuse).
4. Choose the appropriate trigonometric function: Use SOH CAH TOA to select the function that relates the known sides to the unknown side or angle.
5. Solve the equation: Set up the equation and solve for the unknown.
6. Check your answer: Make sure your answer is reasonable in the context of the problem.

Practical Applications

Trigonometry is not just an abstract mathematical concept; it has numerous real-world applications in various fields:

• Navigation: Used in GPS systems and marine navigation to determine positions and directions.
• Engineering: Essential for designing structures, bridges, and machines, ensuring stability and accuracy.
• Physics: Used to analyze motion, forces, and waves.
• Astronomy: Used to measure distances to stars and planets.
• Surveying: Used to measure land and create maps.
• Architecture: Used in building design and ensuring structural integrity.
• Computer Graphics: Used in creating realistic 3D models and animations.

Understanding when to use sin, cos, and tan allows professionals in these fields to solve complex problems and create innovative solutions.

Common Mistakes to Avoid

• Using the wrong function: Carefully identify the known sides and choose the function that relates them to the unknown quantity. Double-check SOH CAH TOA!
• Incorrectly labeling the sides: Make sure you correctly identify the opposite, adjacent, and hypotenuse sides relative to the angle you are using. The opposite and adjacent sides change depending on which acute angle you're focusing on.
• Forgetting to use the inverse trigonometric functions: When finding an angle, remember to use arcsin, arccos, or arctan to get the angle value.
• Using degrees instead of radians (or vice versa): Make sure your calculator is set to the correct mode (degrees or radians) depending on the problem's requirements. This is especially important in calculus and more advanced math.
• Not drawing a diagram: A visual representation can help you understand the problem and avoid mistakes.

Advanced Trigonometry

While sin, cos, and tan are fundamental, trigonometry extends to more advanced concepts:

• Reciprocal Trigonometric Functions: Cosecant (csc), secant (sec), and cotangent (cot) are the reciprocals of sine, cosine, and tangent, respectively. csc(θ) = 1/sin(θ), sec(θ) = 1/cos(θ), cot(θ) = 1/tan(θ). These are used less frequently but are still important.
• Trigonometric Identities: Equations that are true for all values of the variables involved (e.g., the Pythagorean identity: sin²(θ) + cos²(θ) = 1). These are crucial for simplifying trigonometric expressions and solving equations.
• Law of Sines and Law of Cosines: Used to solve non-right triangles. The Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is constant. The Law of Cosines is a generalization of the Pythagorean theorem.
• Radian Measure: An alternative unit for measuring angles, where one radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle.
• Unit Circle: A circle with a radius of 1, used to visualize trigonometric functions for all angles, including those greater than 90 degrees.

Conclusion

Mastering the use of sin, cos, and tan is essential for anyone working with angles, triangles, and periodic phenomena. By understanding the relationships between the sides and angles of a right triangle and by practicing applying these functions to various problems, you can unlock a powerful set of tools for solving real-world challenges in diverse fields. Remember SOH CAH TOA, draw diagrams, and always double-check your work to ensure accurate results. With practice, you'll confidently navigate the world of trigonometry.