Trigonometry of Right and Oblique Triangles · Term 3

Right Triangle Trigonometry

Applying Sine, Cosine, and Tangent ratios to solve for missing components in right triangles.

About This Topic

The Sine and Cosine Laws extend trigonometric principles to non-right (oblique) triangles. In the Ontario curriculum, these laws allow students to solve for any triangle, provided they have enough information. This is a significant leap in mathematical power, as most real world triangles do not contain a perfect 90 degree angle. These laws are essential for advanced studies in physics and geography.

In a Canadian context, these laws are used in surveying the vast and varied landscapes of the country, from the Rockies to the Atlantic coast. They are also vital for search and rescue operations that must triangulate positions across large distances. This topic comes alive when students can physically model the patterns through collaborative investigations of complex geometric problems and real world mapping scenarios.

Learning Objectives

• Calculate the length of an unknown side in a right triangle using sine, cosine, or tangent ratios.
• Determine the measure of an unknown angle in a right triangle using inverse trigonometric functions.
• Apply trigonometric ratios to solve real-world problems involving angles of elevation and depression.
• Explain why trigonometric ratios are constant for similar right triangles.
• Compare the utility of sine, cosine, and tangent in solving different right triangle problems.

Before You Start

Pythagorean Theorem

Why: Students need to be familiar with the relationship between the sides of a right triangle before learning about the ratios of sides.

Properties of Triangles

Why: Understanding that the sum of angles in a triangle is 180 degrees and identifying different types of angles is foundational.

Basic Geometric Definitions

Why: Students must be able to identify and define terms like 'opposite side', 'adjacent side', and 'hypotenuse' in relation to an angle.

Key Vocabulary

Sine (sin)	The ratio of the length of the side opposite an acute angle to the length of the hypotenuse in a right triangle.
Cosine (cos)	The ratio of the length of the adjacent side to an acute angle to the length of the hypotenuse in a right triangle.
Tangent (tan)	The ratio of the length of the side opposite an acute angle to the length of the adjacent side in a right triangle.
Angle of Elevation	The angle measured upward from the horizontal line to the line of sight to an object above the horizontal.
Angle of Depression	The angle measured downward from the horizontal line to the line of sight to an object below the horizontal.

Active Learning Ideas

Inquiry Circle: Triangulating a Signal

Groups are given the positions of two 'towers' and the angles they receive from a 'lost hiker'. They must use the Sine Law to determine the hiker's exact location on a map.

⏱ 50 min·Small Groups

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Formal Debate: Sine Law vs. Cosine Law

Students are presented with various 'oblique' scenarios. They must work in teams to determine which law is applicable (based on SSS, SAS, ASA, or AAS) and defend their choice to the class.

⏱ 35 min·Small Groups

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Gallery Walk: The Law of Cosines in Design

Students find or create a design that uses non-right triangles (like a roof truss or a bridge support). They use the Cosine Law to calculate the necessary lengths and angles and display their work.

⏱ 45 min·Individual

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Real-World Connections

Architects and engineers use trigonometry to calculate roof pitches, ramp slopes, and the heights of buildings or bridges, ensuring structural integrity and accessibility.

Surveyors in Canada's vast terrain use trigonometric principles to map land boundaries, determine elevations, and plan construction projects, even in challenging geographical areas.

Pilots and air traffic controllers utilize trigonometry to calculate flight paths, altitudes, and distances, especially when navigating around obstacles or during landing procedures.

Watch Out for These Misconceptions

Common Misconception

Trying to use the Sine Law when they only have three side lengths (SSS).

What to Teach Instead

Students often default to the Sine Law because it is simpler. Use a think-pair-share to show that without at least one known angle, the Sine Law leaves you with two unknowns, proving the necessity of the Cosine Law for SSS cases.

Common Misconception

Incorrectly applying the order of operations in the Cosine Law.

What to Teach Instead

Students often subtract before multiplying in the '2bc cosA' part of the formula. Peer review of step by step calculations on whiteboards can help students catch and correct this common algebraic error.

Assessment Ideas

Quick Check

Present students with three right triangles, each missing one side or angle. Ask them to write down the trigonometric ratio (sin, cos, or tan) they would use to find the missing component and why. For example, 'For triangle A, I would use tan because I have the opposite and adjacent sides.'

Exit Ticket

Provide students with a scenario: 'A ladder 10 meters long leans against a wall, reaching a height of 8 meters. Calculate the angle the ladder makes with the ground.' Students must show their work using the appropriate trigonometric function and inverse function.

Discussion Prompt

Pose the question: 'Imagine you are standing 50 meters from a tall tree. You measure the angle of elevation to the top of the tree as 30 degrees. How would you determine the height of the tree? Which trigonometric ratio is most helpful here, and why?'

Frequently Asked Questions

When should I use the Sine Law versus the Cosine Law?

Use the Sine Law when you have an 'opposite pair' (an angle and its opposite side) plus one other piece of information. Use the Cosine Law when you have two sides and the angle between them (SAS) or when you have all three sides but no angles (SSS).

How can active learning help students understand the Sine and Cosine Laws?

Active learning, such as the 'Triangulating a Signal' activity, gives students a concrete reason to use these more complex formulas. By working through a simulated rescue mission, students see that these laws aren't just for solving triangles on paper; they are essential for solving problems in space and navigation where right angles are rare.

Is the Cosine Law related to the Pythagorean theorem?

Yes! The Cosine Law is actually a generalized version of the Pythagorean theorem. If you use the Cosine Law on a 90 degree angle, the 'minus 2bc cosA' part becomes zero (because cos 90 is 0), leaving you with exactly a squared + b squared = c squared.

What is an 'oblique' triangle?

An oblique triangle is simply any triangle that does not have a right angle (90 degrees). It can be acute (all angles less than 90) or obtuse (one angle greater than 90). The Sine and Cosine Laws are designed specifically to solve these types of triangles.