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Lesson 5 of 5

Introduction to Trigonometry · Lesson 5 of 5

Chapter Summary and Practice

“Bring ratios, special angles and identities together for one final triangle-powered workout.”

Learning Objectives

• By the end of this lesson, you should be able to. • Recall all six trigonometric ratios and their relationships. • Use standard-angle values without confusion. • Select the correct identity or ratio for a problem. • Solve mixed chapter questions confidently. • Check your readiness before moving to applications of trigonometry.

This lesson brings together the essential ideas from the chapter. Read the summary quickly, study the method-choice table, work through the guided examples and then attempt the practice set without looking back.

Trigonometric Ratios

For a chosen acute angle in a right triangle, the three side names are opposite, adjacent and hypotenuse. The hypotenuse is fixed, while opposite and adjacent depend on the reference angle.

RatioDefinition
sin Aopposite / hypotenuse
cos Aadjacent / hypotenuse
tan Aopposite / adjacent
cosec Ahypotenuse / opposite
sec Ahypotenuse / adjacent
cot Aadjacent / opposite
Important quotient relationshipLaTeX

Specific Angles

The angles 0°, 30°, 45°, 60° and 90° have exact trigonometric values. These values come from special right triangles and the limiting behaviour near 0° and 90°.

Ratio0°30°45°60°90°
sin01/21/√2√3/21
cos1√3/21/√21/20
tan01/√31√3Not defined
cosecNot defined2√22/√31
sec12/√3√22Not defined
cotNot defined√311/√30

Trigonometric Identities

IdentityLaTeX
IdentityLaTeX
IdentityLaTeX

How to Choose the Right Idea

What you are given / askedBest starting idea
Sides of a right triangleUse definitions of sin, cos and tan
One trig ratio, find the othersBuild a triangle or use identities
30°, 45° or 60° appearsUse the standard-angle table
0° or 90° appearsCheck for undefined ratios
Squares such as sin²A and cos²AUse a fundamental identity
Identity proof with sec, cosec, tan or cotConsider rewriting in sin and cos

Common Mistakes

Watch Out For These

• Choosing opposite and adjacent before deciding the reference angle. • Forgetting that tan 90°, sec 90°, cot 0° and cosec 0° are undefined. • Treating sin A as sin × A. • Confusing reciprocal identities with Pythagorean identities. • Trying to prove an identity by checking only one angle. • Forgetting to use the positive square root for side lengths in an acute right triangle.

Guided Practice

Worked Example: Ratios from Sides

Problem
In right triangle ABC, right-angled at B, AB = 9 cm and BC = 12 cm. Find sin A, cos A and tan A.

  1. 1.AC = √(9² + 12²) = 15 cm.
  2. 2.Relative to angle A: opposite = 12, adjacent = 9, hypotenuse = 15.
  3. 3.sin A = 12/15 = 4/5.
  4. 4.cos A = 9/15 = 3/5.
  5. 5.tan A = 12/9 = 4/3.
Worked Example: One Ratio Given

Problem
If sec θ = 13/12, find sin θ and tan θ.

  1. 1.sec θ = hypotenuse/adjacent = 13/12.
  2. 2.Take hypotenuse = 13k and adjacent = 12k.
  3. 3.Opposite = √[(13k)² - (12k)²] = 5k.
  4. 4.Therefore sin θ = 5/13 and tan θ = 5/12.
Worked Example: Standard Angle

Problem
Evaluate 2 sin 30° cos 60° + tan 45°.

  1. 1.= 2(1/2)(1/2) + 1
  2. 2.= 1/2 + 1
  3. 3.= 3/2.
Worked Example: Identity

Problem
If cos A = 12/13, find sin A for acute A.

  1. 1.Use sin²A + cos²A = 1.
  2. 2.sin²A = 1 - (12/13)² = 1 - 144/169 = 25/169.
  3. 3.Since A is acute, sin A is positive.
  4. 4.Therefore sin A = 5/13.
Worked Example: Prove an Identity

Problem
Prove that (1 - sin²A)/cos²A = 1.

  1. 1.From sin²A + cos²A = 1, we have 1 - sin²A = cos²A.
  2. 2.Therefore LHS = cos²A/cos²A = 1.
  3. 3.Hence LHS = RHS.

Quiz

Quick check

Which side is always opposite the 90° angle?

Quick check

What is sin 30°?

Quick check

Which identity is correct?

Quick check

Which ratio is not defined at 90°?

Quick check

If tan A = 1 for an acute angle A, then A is:

Before You Finish the Chapter

Ask yourself: • Can I identify opposite, adjacent and hypotenuse without guessing? • Can I write all six trigonometric ratios from memory? • Can I recall the values at 0°, 30°, 45°, 60° and 90°? • Can I use the three fundamental identities correctly? • Can I prove a simple identity without changing both sides randomly?

Practice Problems

Practice Questions
  1. In △ABC, right-angled at B, AB = 24 cm and BC = 10 cm. Determine (i) sin A, cos A (ii) sin C, cos C.
  2. If sin A = 8/17, calculate cos A and tan A.
  3. Given 20 cot A = 21, find sin A and sec A.
  4. Given sec θ = 25/24, calculate all the other trigonometric ratios.
  5. If angles A and B are acute and cos A = cos B, show that A = B.
  6. If cot θ = 5/12, evaluate (i) [(1 + sin θ)(1 - sin θ)] / [(1 + cos θ)(1 - cos θ)] (ii) cot²θ.
  7. Evaluate: sin 60° cos 30° + sin 30° cos 60°.
  8. Evaluate: 2 tan²45° + cos²30° - sin²60°.
  9. If tan(A + B) = √3 and tan(A - B) = 1/√3, with 0° < A + B ≤ 90° and A > B, find A and B.
  10. State true or false with reason: sin(A + B) = sin A + sin B.
  11. Express sin A, sec A and tan A in terms of cot A.
  12. Write all other trigonometric ratios of A in terms of sec A.
  13. Prove: (cosec θ - cot θ)² = (1 - cos θ)/(1 + cos θ).
  14. Prove: (1 + sin A)/cos A + cos A/(1 + sin A) = 2 sec A.
  15. Prove: (sin A + cosec A)² + (cos A + sec A)² = 7 + tan²A + cot²A.

Key Takeaways

Key Takeaways

• Trigonometry turns side-angle relationships in right triangles into six useful ratios. • Standard-angle values are derived from special triangles and limiting cases. • The three fundamental identities come from Pythagoras theorem. • Choosing the correct reference angle and formula is more important than memorising steps. • A strong chapter-level understanding means you can move between sides, ratios, standard values and identities.

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Trigonometric Identities

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