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Trigonometric Identities Practice Questions with Detailed Solutions – Level 1

Beginner practice on trigonometric identities: Pythagorean identities, complementary angles, standard values and the maximum and minimum of sine-cosine expressions, with worked solutions.

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Questions, answer key and solutions.

Questions
50
Duration
50 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 50-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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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. 50 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 1 Practice Quiz: Basic Identities and Values

This Level 1 set builds the core skill of applying the three Pythagorean identities, complementary-angle relations and standard angle values to simplify trigonometric expressions. It is a beginner practice set on trigonometric identities for competitive exam aspirants, prepared by Aptitude360. Every question comes with a detailed written solution.

Topics covered in this trigonometric identities set

  • Finding the other ratios of an acute angle when one ratio is given
  • Expressions built on the sine-cosine, secant-tangent and cosecant-cotangent identities
  • Complementary angles, such as sine of one angle divided by cosine of its complement
  • Standard values of sine, cosine and tangent at common angles
  • Problems where secant plus tangent or cosecant minus cotangent is given
  • Finding sine times cosine when their sum is known
  • Maximum and minimum of a sine plus cosine combination
  • Products of tangents or cotangents of complementary angle pairs

Key trigonometric formulas for Level 1

  • Pythagorean identity: \( \sin^{2}\theta+\cos^{2}\theta=1 \)
  • Secant form: \( \sec^{2}\theta-\tan^{2}\theta=1 \)
  • Cosecant form: \( \operatorname{cosec}^{2}\theta-\cot^{2}\theta=1 \)
  • Complementary angle: \( \sin(90^{\circ}-\theta)=\cos\theta \)
  • Paired sum and difference: \( (\sec\theta+\tan\theta)(\sec\theta-\tan\theta)=1 \)
  • Range: \( |a\sin\theta+b\cos\theta|\le\sqrt{a^{2}+b^{2}} \)

Revision tips for basic trigonometric identities

  • Memorise Pythagorean triplets so a missing side of a right triangle appears instantly
  • When two angles add up to a right angle, convert one ratio into its co-ratio before simplifying
  • If secant plus tangent is given, use the reciprocal of that value as secant minus tangent
  • Square the sum of sine and cosine to reach their product directly

Who should take this Level 1 set

Beginners in Quant geometry and trigonometry preparing for SSC, banking, CAT or IPMAT should start here before moving to harder levels.

FAQs about Trigonometric Identities Practice Questions with Detailed Solutions – Level 1

What are the three basic trigonometric identities?
Sine squared plus cosine squared equals one, secant squared minus tangent squared equals one, and cosecant squared minus cotangent squared equals one. All three come from the Pythagoras theorem applied to a right triangle.
What is the trigonometric identity for complementary angles?
Sine of an angle equals cosine of its complement, and tangent equals cotangent of its complement. The same pairing holds for secant and cosecant.
How do you find the maximum value of a sin theta plus b cos theta?
The maximum value is the square root of a squared plus b squared. The minimum is minus that same square root.
What is the value of sin 30, cos 60 and tan 45?
Sine of thirty degrees and cosine of sixty degrees are both one half, and tangent of forty-five degrees equals one.

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 15050 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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