Mathematics
Grade 11
15 min
Solve a right triangle
Solve a right triangle
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1
Introduction & Learning Objectives
Learning Objectives
Identify the known and unknown sides and angles of a right triangle.
Select the appropriate trigonometric ratio (sine, cosine, tangent) to find a missing side.
Use inverse trigonometric functions to find a missing angle.
Apply the Pythagorean theorem to find a missing side when two other sides are known.
Solve a right triangle completely by finding all three unknown parts (sides and/or angles).
Apply the process of solving a right triangle to real-world application problems involving angles of elevation and depression.
How can an architect determine the height of a skyscraper without ever leaving the ground? 📐 The answer lies in solving a simple triangle!
This tutorial will guide you through the process of 'solving a right triangle,' which mea...
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Key Concepts & Vocabulary
TermDefinitionExample
Solving a Right TriangleThe process of finding the measures of all unknown sides and all unknown angles of a right triangle, given at least one side length and one other piece of information (another side or an acute angle).Given a right triangle with one angle of 30° and the adjacent side of 10 cm, 'solving' it means finding the length of the other two sides and the measure of the third angle (60°).
Trigonometric Ratios (SOH CAH TOA)Ratios of the lengths of two sides of a right triangle, related to one of its acute angles. SOH: Sine = Opposite/Hypotenuse, CAH: Cosine = Adjacent/Hypotenuse, TOA: Tangent = Opposite/Adjacent.In a right triangle with an angle θ, if the side opposite is 3 and the hypotenuse is 5, then sin(θ) = 3/5.
Inverse Trigonometric Functio...
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Core Formulas
Primary Trigonometric Ratios (SOH CAH TOA)
\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \quad | \quad \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \quad | \quad \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}
Use these to find a missing side length when you know one side and one acute angle. Choose the ratio that connects the known angle, the known side, and the unknown side you want to find.
Pythagorean Theorem
a^2 + b^2 = c^2
Use this to find the length of a third side of a right triangle when you know the lengths of the other two sides. 'a' and 'b' are the legs, and 'c' is always the hypotenuse.
Inverse Trigonometric Functions
\theta = \sin^{-1}\left(\frac{\text{Opposite}}{\text{Hypotenuse}}\right) \quad |...
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Challenging
From a point on the ground, the angle of elevation to the top of a cliff is 30°. After walking 100 meters closer to the cliff, the angle of elevation is 45°. What is the height of the cliff? Round to two decimal places.
A.236.60 meters
B.136.60 meters
C.73.21 meters
D.173.21 meters
Challenging
In a right triangle, the side opposite angle θ is 'x' and the hypotenuse is '3x'. Which expression correctly represents sin(θ) + cos(θ)?
A.(1 + 2√2) / 3
B.1
C.(1 + √3) / 2
D.Cannot be determined
Challenging
A student is solving a right triangle with legs a = 7 and b = 11. They calculate angle A as tan⁻¹(7/11) ≈ 32.47°. They then calculate angle B as 90° - 32.47° = 57.53°. To avoid premature rounding errors, which of the following is the most accurate value for angle B?
A.57.53°
B.57.52°
C.57.54°
D.tan⁻¹(11/7)
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