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Geometry and Trigonometry - Right-angled triangle trigonometry (SOH CAH TOA)

Grade 9IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A right-angled triangle is defined by having one internal angle of exactly 90∘90^\circ, often represented visually by a small square in the corner. The side opposite this 90∘90^\circ angle is called the hypotenuse, which is always the longest side of the triangle.

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The names of the other two sides depend on their position relative to a chosen reference angle, usually denoted as θ\theta. The 'Opposite' side is the side directly across from θ\theta, while the 'Adjacent' side is the leg that helps form the angle θ\theta and touches the vertex of that angle.

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The acronym SOH CAH TOA is a mnemonic used to remember the three primary trigonometric ratios. SOH stands for sin⁡(θ)=OppositeHypotenuse\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}, CAH stands for cos⁡(θ)=AdjacentHypotenuse\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}, and TOA stands for tan⁡(θ)=OppositeAdjacent\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}.

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To solve for an unknown side when an angle and one other side are known, you select the ratio that involves the two sides you are working with (the known and the unknown) and rearrange the equation to isolate the variable.

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To find an unknown angle when two sides are known, you must use inverse trigonometric functions, denoted as sin⁡−1\sin^{-1}, cos⁡−1\cos^{-1}, or tan⁡−1\tan^{-1} on a calculator. This effectively 'undoes' the trigonometric ratio to reveal the angle measurement in degrees.

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The Angle of Elevation is the upward angle measured from a horizontal line of sight to an object. Visually, this looks like a line starting at a point and sloping upwards to the right or left. The Angle of Depression is the downward angle from a horizontal line of sight to an object below.

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The Pythagorean Theorem (a2+b2=c2a^2 + b^2 = c^2) is intrinsically linked to trigonometry in right-angled triangles. It allows you to calculate a third side length if any two other sides are known, providing a way to verify trigonometric calculations.

📐Formulae

sin⁡(θ)=OppositeHypotenuse\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}

cos⁡(θ)=AdjacentHypotenuse\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}

tan⁡(θ)=OppositeAdjacent\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}

θ=sin⁡−1(OppositeHypotenuse)\theta = \sin^{-1}\left(\frac{\text{Opposite}}{\text{Hypotenuse}}\right)

θ=cos⁡−1(AdjacentHypotenuse)\theta = \cos^{-1}\left(\frac{\text{Adjacent}}{\text{Hypotenuse}}\right)

θ=tan⁡−1(OppositeAdjacent)\theta = \tan^{-1}\left(\frac{\text{Opposite}}{\text{Adjacent}}\right)

a2+b2=c2a^2 + b^2 = c^2 (Pythagorean Theorem)

💡Examples

Problem 1:

In a right-angled triangle, the hypotenuse is 1212 cm long and one of the acute angles is 35∘35^\circ. Find the length of the side opposite to the 35∘35^\circ angle, correct to 2 decimal places.

Solution:

  1. Identify the given information: Hypotenuse =12= 12, θ=35∘\theta = 35^\circ, and we need to find the Opposite side (xx).
  2. Choose the correct ratio: Since we have the Hypotenuse and want the Opposite, we use SOH: sin⁡(θ)=OppositeHypotenuse\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}.
  3. Set up the equation: sin⁡(35∘)=x12\sin(35^\circ) = \frac{x}{12}.
  4. Rearrange to solve for xx: x=12×sin⁡(35∘)x = 12 \times \sin(35^\circ).
  5. Calculate: x≈12×0.5736≈6.88x \approx 12 \times 0.5736 \approx 6.88 cm.

Explanation:

We use the Sine ratio because the problem involves the hypotenuse and the side opposite the given angle. Multiplying the hypotenuse by the sine of the angle isolates the unknown side length.

Problem 2:

A ladder is leaning against a wall. The foot of the ladder is 33 m away from the base of the wall, and the ladder reaches 77 m up the wall. Calculate the angle that the ladder makes with the ground.

Solution:

  1. Identify the given information: The distance from the wall is the Adjacent side (33 m) and the height up the wall is the Opposite side (77 m). We need to find the angle θ\theta.
  2. Choose the correct ratio: Since we have Opposite and Adjacent, we use TOA: tan⁡(θ)=OppositeAdjacent\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}.
  3. Set up the equation: tan⁡(θ)=73\tan(\theta) = \frac{7}{3}.
  4. Use the inverse tangent function to find θ\theta: θ=tan⁡−1(73)\theta = \tan^{-1}\left(\frac{7}{3}\right).
  5. Calculate: θ≈tan⁡−1(2.333)≈66.8∘\theta \approx \tan^{-1}(2.333) \approx 66.8^\circ.

Explanation:

Because we know the two legs of the triangle (opposite and adjacent) but not the hypotenuse, the tangent ratio is the most direct way to find the angle. We use the inverse tangent function to convert the ratio of the sides back into an angle measurement.