Use the three trigonometric identities
Apply the supplied Pythagorean identities to find missing trigonometric values, simplify expressions and check signs from the quadrant.
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which identity is valid when cos θ ≠ 0?
What you will learn
- Use sin²θ + cos²θ = 1 and the identities for secant, tangent, cosecant and cotangent
- Rearrange an identity to find a missing value
- Choose the sign of a square root using the angle's quadrant and state domain restrictions
The three identities come from the unit circle
The unit-circle equation is cos²θ + sin²θ = 1. Dividing by cos²θ (where cos θ ≠ 0) gives 1 + tan²θ = sec²θ.
Dividing the same equation by sin²θ (where sin θ ≠ 0) gives 1 + cot²θ = cosec²θ. These three identities are supplied in the Cambridge formula list, but students must choose and use them correctly.
An identity gives a magnitude; the quadrant gives its sign
If tan θ = 3/4, then sec²θ = 1 + 9/16 = 25/16, so sec θ is either 5/4 or −5/4. The quadrant decides which sign applies.
Keep restrictions visible. The tangent-secant identity requires cos θ ≠ 0; the cotangent-cosecant identity requires sin θ ≠ 0. If you divide by an expression that could be zero, consider that case separately.
Use identities to replace one family of functions with another before simplifying. For example, sec²θ − 1 can be replaced by tan²θ, which may expose a common factor or a simpler equation.
Pause and explain
If cosec²θ = 25/9, what are the possible values of cot²θ?
Worked example
Given tan θ = −3/4 and θ is in Quadrant IV, find sec θ, sin θ, cos θ, cot θ and cosec θ.
Show the worked solution
- In Quadrant IV, cosine is positive and sine is negative, so sec θ must be positive.
- sec²θ = 1 + tan²θ = 1 + 9/16 = 25/16; hence sec θ = 5/4 and cos θ = 4/5.
- sin θ = tan θ × cos θ = −3/5. Taking reciprocals gives cot θ = −4/3 and cosec θ = −5/3.
Answer sec θ = 5/4, sin θ = −3/5, cos θ = 4/5, cot θ = −4/3 and cosec θ = −5/3.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Given tan θ = 3/4 in Quadrant I, find sec θ, sin θ, cos θ, cot θ and cosec θ.
Give me a hint
Use sec²θ = 1 + tan²θ; choose positive signs in Quadrant I.
Compare my reasoning
- sec²θ = 1 + 9/16 = 25/16, so sec θ = 5/4 and cos θ = 4/5.
- sin θ = tan θ × cos θ = 3/5.
- Then cot θ = 4/3 and cosec θ = 5/3.
sec θ = 5/4, sin θ = 3/5, cos θ = 4/5, cot θ = 4/3 and cosec θ = 5/3.
Look for these in your work
- Identity rearranged correctly
- Quadrant signs used
- All reciprocal values checked
Given sin θ = −5/13 and θ is in Quadrant IV, find the other five trigonometric values.
Give me a hint
Cosine is positive in Quadrant IV; use sin²θ + cos²θ = 1.
Compare my reasoning
- cos²θ = 1 − 25/169 = 144/169, so cos θ = 12/13.
- tan θ = (−5/13)/(12/13) = −5/12 and cot θ = −12/5.
- sec θ = 13/12 and cosec θ = −13/5.
cos θ = 12/13, tan θ = −5/12, cot θ = −12/5, sec θ = 13/12 and cosec θ = −13/5.
Look for these in your work
- Square root sign matches the quadrant
- Ratios simplified
- Reciprocals retain signs
Simplify (sec²θ − 1)/tan θ and state when the simplification is valid.
Give me a hint
Replace sec²θ − 1 using an identity, then consider the denominator.
Compare my reasoning
- Use sec²θ − 1 = tan²θ.
- For tan θ ≠ 0, tan²θ/tan θ = tan θ.
- The expression also requires cos θ ≠ 0 for sec and tan to be defined.
The expression simplifies to tan θ when cos θ ≠ 0 and tan θ ≠ 0 (equivalently sin θ ≠ 0 and cos θ ≠ 0).
Look for these in your work
- Correct identity selected
- Cancellation condition stated
- Original domain restrictions retained
Common mistakes
- Forgetting the ± when taking a square root
- Choosing a sign without checking the quadrant
- Dividing by sine or cosine without checking whether it is zero
- Treating a supplied identity as a formula that holds outside its domain
What can you explain now?
If tan θ = 3/4 and θ is in Quadrant I, what is sec θ?
Compare with the explanation
5/4
sec²θ = 1 + tan²θ = 25/16 and secant is positive in Quadrant I.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
State the three supplied trigonometric identities and explain how the quadrant determines the sign after taking a square root.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.