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    C3 Trigonometry D - Questions

    Uploaded by

    Benny Nguyen
    This document contains 17 problems involving trigonometric identities. The problems cover topics such as expressing trigonometric functions in terms of sine and cosine, using addition and subtraction formulas, solving trigonometric equations, and proving trigonometric identities. Some problems involve finding exact values of trigonometric functions without using a calculator.

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    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd

    C3 Trigonometry D - Questions

    Uploaded by

    Benny Nguyen
    201 views2 pages
    This document contains 17 problems involving trigonometric identities. The problems cover topics such as expressing trigonometric functions in terms of sine and cosine, using addition and subtraction formulas, solving trigonometric equations, and proving trigonometric identities. Some problems involve finding exact values of trigonometric functions without using a calculator.

    Original Description:

    trigonometry core 3 worksheet D by physicsandmathtutors.com

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    This document contains 17 problems involving trigonometric identities. The problems cover topics such as expressing trigonometric functions in terms of sine and cosine, using addition and subtraction formulas, solving trigonometric equations, and proving trigonometric identities. Some problems involve finding exact values of trigonometric functions without using a calculator.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    201 views2 pages

    C3 Trigonometry D - Questions

    Uploaded by

    Benny Nguyen
    This document contains 17 problems involving trigonometric identities. The problems cover topics such as expressing trigonometric functions in terms of sine and cosine, using addition and subtraction formulas, solving trigonometric equations, and proving trigonometric identities. Some problems involve finding exact values of trigonometric functions without using a calculator.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 2

    PMT

    C3 TRIGONOMETRY Worksheet D

    1 a Write down the identities for sin (A + B) and cos (A + B).


    b Use these identities to obtain similar identities for sin (A − B) and cos (A − B).
    c Use the above identities to obtain similar identities for tan (A + B) and tan (A − B).

    2 Express each of the following in the form sin α, where α is acute.


    a sin 10° cos 30° + cos 10° sin 30° b sin 67° cos 18° − cos 67° sin 18°
    c sin 62° cos 74° + cos 62° sin 74° d cos 14° cos 39° − sin 14° sin 39°

    3 Express as a single trigonometric ratio


    a cos A cos 2A − sin A sin 2A b sin 4A cos B − cos 4A sin B
    tan 2 A + tan 5 A
    c d cos A cos 3A + sin A sin 3A
    1 − tan 2 A tan 5 A

    4 Find in exact form, with a rational denominator, the value of


    a sin 15° b sin 165° c cosec 15° d cos 75°
    e cos 15° f sec 195° g tan 75° h cosec 105°

    5 Find the maximum value that each expression can take and the smallest positive value of x, in
    degrees, for which this maximum occurs.
    a cos x cos 30° + sin x sin 30° b 3 sin x cos 45° + 3 cos x sin 45°
    c sin x cos 67° − cos x sin 67° d 4 sin x sin 108° − 4 cos x cos 108°

    6 Find the minimum value that each expression can take and the smallest positive value of x, in
    radians in terms of π, for which this minimum occurs.
    π π π π
    a sin x cos 3
    − cos x sin 3
    b 2 cos x cos 6
    − 2 sin x sin 6

    c cos 4x cos x + sin 4x sin x d 6 sin 2x cos 3x − 6 sin 3x cos 2x

    7 Given that sin A = 45 , 0 < A < 90° and that cos B = 2


    3
    , 0 < B < 90°, find without using a
    calculator the value of
    a tan A b sin B c cos (A + B) d sin (A + B)

    8 Given that cosec C = 53 , 0 < C < 90° and that sin D = 5


    13
    , 90° < D < 180°, find without using
    a calculator the value of
    a cos C b cos D c sin (C − D) d sec (C − D)

    9 Solve each equation for θ in the interval 0 ≤ θ ≤ 360.


    Give your answers to 1 decimal place where appropriate.
    tan 2θ ° − tan 60°
    a sin θ ° cos 15° + cos θ ° sin 15° = 0.4 b =1
    1 + tan 2θ ° tan 60°
    c cos (θ − 60)° = sin θ ° d 2 sin θ ° + sin (θ + 45)° = 0
    e sin (θ + 30)° = cos (θ − 45)° f 3 cos (2θ + 60)° − sin (2θ − 30)° = 0

     Solomon Press
    PMT

    C3 TRIGONOMETRY Worksheet D continued

    10 Find the value of k such that for all real values of x


    π π
    cos (x + 3
    ) − cos (x − 3
    ) ≡ k sin x.

    11 Prove each identity.


    π π
    a cos x − cos (x − 3
    ) ≡ cos (x + 3
    )
    π π
    b sin (x − 6
    ) + cos x ≡ sin (x + 6
    )

    12 a Use the identity for sin (A + B) to express sin 2A in terms of sin A and cos A.
    b Use the identity for cos (A + B) to express cos 2A in terms of sin A and cos A.
    c Hence, express cos 2A in terms of
    i cos A ii sin A
    d Use the identity for tan (A + B) to express tan 2A in terms of tan A.

    13 Solve each equation for x in the interval 0 ≤ x ≤ 360°.


    Give your answers to 1 decimal place where appropriate.
    a cos 2x + cos x = 0 b sin 2x + cos x = 0
    c 2 cos 2x = 7 sin x d 11 cos x = 4 + 3 cos 2x
    e tan 2x − tan x = 0 f sec x − 4 sin x = 0
    g 5 sin 4x = 2 sin 2x h 2 sin2 x − cos 2x − cos x = 0

    14 Prove each identity.


    a (cos x + sin x)2 ≡ 1 + sin 2x b tan x (1 + cos 2x) ≡ sin 2x
    2sin x
    c ≡ tan 2x d tan x + cot x ≡ 2 cosec 2x
    2cos x − sec x
    e cosec 2x − cot 2x ≡ tan x f (cos x + sin x)(cosec x − sec x) ≡ 2 cot 2x
    1 − sin 2 x
    g ≡ sin x h cos 3x ≡ 4 cos3 x − 3 cos x
    cosec x − 2cos x

    15 Use the double angle identities to prove that


    x x
    a cos x ≡ 2 cos2 −1 b sin2 ≡ 1
    2
    (1 − cos x)
    2 2

    7 A
    16 a Given that cos A = 9
    , 0 < A < 90°, find the exact value of sin without using a calculator.
    2
    B
    b Given that cos B = − 83 , 90° < B < 180°, find the value of cos , giving your answer in the
    2
    form k 5 .

    17 Prove each identity.


    2 x 1 + cos x x
    a ≡ sec2 b ≡ cot2
    1 + cos x 2 1 − cos x 2

     Solomon Press

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