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B.Sc Mathematics Trigonometric Identities and Equations — practice questions — page 2

Explore 73 free Trigonometric Identities and Equations questions in B.Sc Mathematics Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Trigonometric Identities and Equations questions and exercises with answers

Showing questions 31–60 of 73.

Which expression is not one of the three standard forms of cos(2x)?Read the answer and worked solution →Which quotient is the double-angle formula for tangent?Read the answer and worked solution →Given tan x = -3/4 with x in quadrant II, find sin(2x).Read the answer and worked solution →For the same angle with tan x = -3/4 in quadrant II, what is cos(2x)?Read the answer and worked solution →Substituting tan x = -3/4 into the double-angle formula for tangent gives which value for tan(2x)?Read the answer and worked solution →Write cos(6x) in terms of cos(3x) using a double-angle formula.Read the answer and worked solution →Expanding (sin x + cos x)^2 and applying a Pythagorean identity yields which expression?Read the answer and worked solution →What do the reduction formulas accomplish?Read the answer and worked solution →Which expression equals sin^2 x by the reduction formula?Read the answer and worked solution →Applying the reduction formula twice, cos^4 x equals which expression?Read the answer and worked solution →Using the half-angle formula for sine with cos 30 degrees = sqrt(3)/2, find sin 15 degrees.Read the answer and worked solution →What do the product-to-sum formulas do?Read the answer and worked solution →The formula for cos a cos b is obtained by combining the two cosine formulas in which way?Read the answer and worked solution →How is the product-to-sum formula for sin a sin b derived?Read the answer and worked solution →Which expression equals cos a cos b?Read the answer and worked solution →Which expression equals sin a cos b?Read the answer and worked solution →Write 2 cos(7x/2) cos(3x/2) as a sum.Read the answer and worked solution →Express sin(4t) cos(2t) as a sum containing no products.Read the answer and worked solution →How are the sum-to-product formulas obtained?Read the answer and worked solution →Which product equals sin a + sin b?Read the answer and worked solution →Which product equals cos a - cos b?Read the answer and worked solution →Write sin(4t) - sin(2t) as a product.Read the answer and worked solution →Evaluate cos 15 degrees minus cos 75 degrees using the sum-to-product formula.Read the answer and worked solution →Simplify (cos(4t) - cos(2t)) / (sin(4t) + sin(2t)) with the sum-to-product formulas.Read the answer and worked solution →Who is named as the founder of geometry, credited with measuring the Great Pyramid of Giza by its shadow?Read the answer and worked solution →What distinguishes a trigonometric identity from a trigonometric equation?Read the answer and worked solution →The domain of the tangent function excludes which set of angles?Read the answer and worked solution →On the interval from 0 to 2 pi, which angles satisfy cos t = 1/2?Read the answer and worked solution →Which two angles in one full turn satisfy sin t = 1/2?Read the answer and worked solution →Solve 2 cos t - 3 = -5 exactly on the interval from 0 to 2 pi.Read the answer and worked solution →

Worked practice questions

Which expression is not one of the three standard forms of cos(2x)?

  1. cos^2 x - sin^2 x
  2. 2 cos^2 x - 1
  3. 1 - 2 sin^2 x
  4. 2 sin^2 x - 1
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Correct answer: D. 2 sin^2 x - 1

1. Setting both angles equal in the cosine sum formula gives cos(2x) = cos^2 x - sin^2 x. 2. Replacing cos^2 x by 1 - sin^2 x turns that into 1 - 2 sin^2 x. 3. Replacing sin^2 x by 1 - cos^2 x turns it into 2 cos^2 x - 1. 4. Those are the three standard forms, so 2 sin^2 x - 1 is the odd one out. 5. In fact 2 sin^2 x - 1 is the negative of cos(2x), so a student who picks it has flipped a sign. _Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 9 "Trigonometric Identities and Equations", section 9.3 Double-Angle, Half-Angle, and Reduction Formulas_

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Which quotient is the double-angle formula for tangent?

  1. 2 tan x / (1 + tan^2 x)
  2. tan x / (1 - tan^2 x)
  3. 2 tan x / (1 - tan^2 x)
  4. 2 tan x / (tan^2 x - 1)
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Correct answer: C. 2 tan x / (1 - tan^2 x)

1. Set both angles equal in tan(a + b) = (tan a + tan b) / (1 - tan a tan b). 2. The numerator becomes tan x + tan x = 2 tan x. 3. The denominator becomes 1 - tan x times tan x, that is 1 - tan^2 x. 4. So tan(2x) = 2 tan x / (1 - tan^2 x). 5. The version with tan^2 x - 1 in the denominator is the negative of the correct value. _Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 9 "Trigonometric Identities and Equations", section 9.3 Double-Angle, Half-Angle, and Reduction Formulas_

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Given tan x = -3/4 with x in quadrant II, find sin(2x).

  1. 24 / 25
  2. -24 / 25
  3. -7 / 25
  4. 7 / 25
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Correct answer: B. -24 / 25

1. In quadrant II the adjacent side is negative, so take opposite 3 and adjacent -4. 2. The Pythagorean Theorem gives (-4)^2 + 3^2 = 16 + 9 = 25, so the hypotenuse is 5. 3. That makes sin x = 3/5 and cos x = -4/5. 4. Apply sin(2x) = 2 sin x cos x. 5. Substituting gives 2 times (3/5) times (-4/5) = -24/25. 6. The positive 24/25 ignores the negative cosine that quadrant II forces. 7. The value 7/25 is cos(2x) for this angle, so it answers a different part of the problem. _Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 9 "Trigonometric Identities and Equations", section 9.3 Double-Angle, Half-Angle, and Reduction Formulas_

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