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Trigonometric Identities Practice Questions with Detailed Solutions – Level 3

Advanced trigonometric identity problems: co-ratio angle equations, sixth powers, double-angle maxima, parameter substitutions and reciprocal extremes, with full worked solutions.

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Questions, answer key and solutions.

Questions
50
Duration
75 min
Marks
50

Negative marking

Attempts
100
Access
Free
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Normal practice, or a Simulated test

Both use the same questions. Normal practice is untimed and needs no account: choose an option and the answer and solution appear at once. The Simulated test is the real computer-based format, with one 75-minute clock, a question palette and full screen, and your score, the key and the solutions arrive when you submit.

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Untimed

Every question on one page, loading as you scroll. Choose an option and see straight away whether it was right.

  • No timer. Nothing to start and nothing to submit. Stop whenever you like.
  • Answered as you choose. Tap an option and it is marked at once, with the correct answer shown.
  • Solutions in place. The explanation, and the video where one has been added, sit under the question.
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Real exam conditions

Simulated

Timed CBT

The computer-based test format: one clock, a question palette, and a score the moment you submit.

  • A clock that does not pause. 75 minutes from the moment you begin, and the paper is submitted for you when it reaches zero.
  • Question palette. Answered, not answered and marked for review, section by section, as the real interface has it.
  • Full screen, proctored. The test takes over the screen; leaving it three times submits the paper.
  • Answers hidden until you submit. The key, the explanations and your score all arrive together at the end.

About this test

Trigonometric Identities Level 3 Advanced Practice Questions

This set lets a student check whether trigonometric identities hold up in advanced trigonometric identity problems. Level 3 on Aptitude360 covers angle equations from complementary relations, sixth powers, double-angle extremes, parameter substitutions and unusual products, with a full written solution for every question.

Advanced trigonometric identity topics in Level 3

  • Solving for theta when a secant equals a cosecant or a tangent product equals one
  • Product of the maximum and minimum of a shifted sine-cosine expression
  • Greatest value of a reciprocal of a sine-cosine expression
  • Maxima involving the double angle and a cosine of a shifted angle
  • Sixth and fourth powers from a given sum of sine and cosine
  • Expressing cosine or secant plus tangent in terms of a parameter
  • Polynomial expressions in sine powers using a given cosine condition
  • Counting integer values a sine-cosine expression can take

Key formulas for advanced trigonometric identity questions

  • Sixth powers: \( \sin^{6}\theta+\cos^{6}\theta=1-3\sin^{2}\theta\cos^{2}\theta \)
  • Double angle: \( \sin2\theta=2\sin\theta\cos\theta \)
  • Double angle: \( \cos2\theta=\cos^{2}\theta-\sin^{2}\theta \)
  • Parameter form: \( \sec\theta+\tan\theta=m\Rightarrow\cos\theta=\dfrac{2m}{m^{2}+1} \)
  • Acute angles: \( \sin A=\cos B\Rightarrow A+B=90^{\circ} \)
  • Maximum: \( a\sin\theta+b\cos\theta+c\le c+\sqrt{a^{2}+b^{2}} \)

Revision tips for Level 3 trigonometry

  • Convert double-angle terms to the form a sine plus b cosine of the same angle before finding extremes
  • For a reciprocal expression, the greatest value comes from the smallest positive denominator
  • Set the two angles in a co-ratio equation to add up to a right angle, then solve the linear equation
  • Expand a cosine of a shifted angle with the compound-angle formula before combining terms

Who should take Level 3

Aspirants who are confident with the basic and intermediate levels and want final-stage practice on tough identity questions for SSC, CAT, IPMAT and similar exams.

FAQs about Trigonometric Identities Practice Questions with Detailed Solutions – Level 3

What is the formula for sin6 theta plus cos6 theta?
It equals one minus three times sine squared theta times cosine squared theta. It comes from cubing the identity sine squared plus cosine squared equals one.
If sec theta plus tan theta equals m, what is cos theta?
Cos theta equals two m divided by the sum of m squared and one. This uses the fact that secant minus tangent is the reciprocal of m.
If sin A equals cos B, what is A plus B?
When both angles are acute, A plus B equals a right angle, because sine of an angle equals cosine of its complement.
What are the double angle formulas in trigonometry?
Sine of twice an angle equals two sine times cosine, and cosine of twice an angle equals cosine squared minus sine squared. Cosine of twice an angle can also be written as one minus twice sine squared.

Test structure

How the paper is split, and how long each part runs. A section you have finished cannot be returned to.

SectionQuestionsTime
Trigonometric Identities Level 35075 min

Marking & rules

Total marks
50 across 50 questions.
Negative marking
A wrong answer costs marks. Leaving a question unanswered costs nothing.
Attempts
100 in total. An attempt is spent the moment you begin one.

What you get when you submit

The analysis is ready the moment you submit the paper.

Score and accuracy

Marks, percentage, right, wrong and skipped, and the time you took.

Section by section

The same figures for each section, so a weak one cannot hide in an average.

The full answer key

Every question with your answer, the right one, the working, and the video where there is one.

Where you stand

Your score against the spread of everyone else who has sat this test.

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