Maths Tutorial: Trigonometry SOH CAH TOA (trigonometric ratios)

Further Maths

5 chapters7 takeaways10 key terms5 questions

Overview

This video explains the fundamental trigonometric ratios: sine (SOH), cosine (CAH), and tangent (TOA), which are applicable only to right-angled triangles. It details how to correctly label the sides of a right-angled triangle (hypotenuse, opposite, and adjacent) relative to a specific angle. The tutorial demonstrates how to use these ratios to find unknown side lengths or angles within a triangle when sufficient information is provided, emphasizing the importance of calculator mode (degrees vs. radians) and the use of inverse trigonometric functions to solve for angles.

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Chapters

Trigonometric ratios (sine, cosine, tangent) are used in right-angled triangles to find unknown angles or side lengths.
These ratios require a triangle with a 90-degree angle to be applicable.
The three primary ratios are SOH (Sine = Opposite/Hypotenuse), CAH (Cosine = Adjacent/Hypotenuse), and TOA (Tangent = Opposite/Adjacent).

Understanding these basic ratios is the foundation for solving a wide range of geometry and physics problems involving triangles.

The hypotenuse is always the longest side, directly opposite the right angle.
The 'opposite' side is the side directly across from the angle of interest.
The 'adjacent' side is the side next to the angle of interest, which is not the hypotenuse.
The labeling of 'opposite' and 'adjacent' sides changes depending on which angle is chosen as the reference.

Correctly identifying and labeling the sides is crucial for applying the correct trigonometric ratio and obtaining accurate calculations.

If the reference angle is at the bottom of the triangle, the side directly across from it is 'opposite', and the side next to it (not the hypotenuse) is 'adjacent'. If the reference angle is at the top, these labels switch.

Choose the ratio (SOH, CAH, TOA) that uses the known side(s) and the unknown side.
Set up the equation using the chosen ratio, the known angle, and the side lengths.
Rearrange the equation to solve for the unknown side, often involving multiplication.
Ensure the calculator is set to degree mode for calculations involving angles in degrees.

This process allows you to calculate the exact length of a side in a right-angled triangle when you know an angle and another side.

Given a 43° angle and a hypotenuse of 12, to find the opposite side (x), use Sine: sin(43°) = x/12, so x = 12 * sin(43°) ≈ 8.18.

When trying to find an angle, use the inverse trigonometric functions (inverse sine, inverse cosine, inverse tangent).
These are often denoted as sin⁻¹, cos⁻¹, or tan⁻¹ on calculators.
Set up the ratio using the known sides (e.g., opposite/hypotenuse for sine).
Apply the appropriate inverse function to the ratio to find the angle.

This enables you to determine the precise measure of an angle within a right-angled triangle when you know the lengths of two sides.

If the opposite side is 3 and the hypotenuse is 5, the angle (θ) is found using inverse sine: θ = sin⁻¹(3/5) ≈ 37°.

Real-world problems can be modeled using right-angled triangles.
Identify the right angle, the angle of interest, and the relevant sides (opposite, adjacent, hypotenuse).
Use the appropriate trigonometric ratio or inverse function to solve for unknown angles or lengths.
The same trigonometric principles apply whether solving for a side or an angle, or in abstract problems versus practical scenarios.

This demonstrates how trigonometry is a practical tool for solving problems in fields like construction and engineering.

For a flagpole guy rope anchored 8m from the base, with the rope attached 9m up the pole, the angle with the ground is tan⁻¹(9/8) ≈ 48.4°, and the rope length is approximately 12m.

Key takeaways

1Right-angled triangles are essential for applying basic trigonometric ratios (SOH CAH TOA).
2Accurate labeling of triangle sides (opposite, adjacent, hypotenuse) relative to the angle of interest is the first critical step.
3SOH CAH TOA provides direct formulas to relate angles and side ratios.
4To find an unknown side, use the direct trigonometric ratio (sin, cos, tan) and solve for the side.
5To find an unknown angle, use the inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹) after determining the correct side ratio.
6Always ensure your calculator is in the correct mode (degrees or radians) for trigonometric calculations.
7Trigonometry is a powerful tool for solving problems involving unknown lengths and angles in real-world scenarios.

Key terms

Trigonometric RatiosRight-angled TriangleHypotenuseOpposite SideAdjacent SideSine (sin)Cosine (cos)Tangent (tan)SOH CAH TOAInverse Trigonometric Functions (sin⁻¹, cos⁻¹, tan⁻¹)

Test your understanding

1What are the three primary trigonometric ratios, and what does each ratio represent in terms of a right-angled triangle's sides?
2How does the labeling of the 'opposite' and 'adjacent' sides change based on the reference angle within a right-angled triangle?
3Explain the difference between using a direct trigonometric function (like sin) to find a side and using an inverse trigonometric function (like sin⁻¹) to find an angle.
4Why is it important to correctly identify the hypotenuse, opposite, and adjacent sides before applying any trigonometric ratio?
5Describe a situation where you would use the tangent ratio (TOA) versus the sine ratio (SOH) to solve for an unknown.