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Evaluate the following trigonometric expressions: 1.1 sec 30°, 1.2 cos 60° + tan 45°, 1.3 cos 30° × sin 60°, 1.4 sin 60° cos 30° - cos 60° sin 30°

Mathematics

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Here are the solutions for the trigonometric expressions:

Part 1.1· 4 steps

sec⁡30∘\sec 30^\circ

  1. Recall the definition of sec⁡θ\sec \theta.

    sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}

  2. Substitute the value of cos⁡30∘\cos 30^\circ.

    cos⁡30∘=32\cos 30^\circ = \frac{\sqrt{3}}{2}

  3. Calculate sec⁡30∘\sec 30^\circ.

    sec⁡30∘=132=23\sec 30^\circ = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}

  4. Rationalize the denominator.

    sec⁡30∘=23×33=233\sec 30^\circ = \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}

    The value is 233\boxed{\frac{2\sqrt{3}}{3}}.

Part 1.2· 2 steps

cos⁡60∘+tan⁡45∘\cos 60^\circ + \tan 45^\circ

  1. Substitute the known values.

    cos⁡60∘=12\cos 60^\circ = \frac{1}{2} tan⁡45∘=1\tan 45^\circ = 1

  2. Add the values.

    cos⁡60∘+tan⁡45∘=12+1=12+22=32\cos 60^\circ + \tan 45^\circ = \frac{1}{2} + 1 = \frac{1}{2} + \frac{2}{2} = \frac{3}{2}

    The value is 32\boxed{\frac{3}{2}}.

Part 1.3· 2 steps

cos⁡30∘×sin⁡60∘\cos 30^\circ \times \sin 60^\circ

  1. Substitute the known values.

    cos⁡30∘=32\cos 30^\circ = \frac{\sqrt{3}}{2} sin⁡60∘=32\sin 60^\circ = \frac{\sqrt{3}}{2}

  2. Multiply the values.

    cos⁡30∘×sin⁡60∘=32×32=3×32×2=34\cos 30^\circ \times \sin 60^\circ = \frac{\sqrt{3}}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3} \times \sqrt{3}}{2 \times 2} = \frac{3}{4}

    The value is 34\boxed{\frac{3}{4}}.

Part 1.4· 3 steps

sin⁡60∘cos⁡30∘−cos⁡60∘sin⁡30∘\sin 60^\circ \cos 30^\circ - \cos 60^\circ \sin 30^\circ

  1. Recognize the trigonometric identity for sin⁡(A−B)\sin(A-B).

    sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin(A-B) = \sin A \cos B - \cos A \sin B

  2. Apply the identity with A=60∘A=60^\circ and B=30∘B=30^\circ.

    sin⁡60∘cos⁡30∘−cos⁡60∘sin⁡30∘=sin⁡(60∘−30∘)=sin⁡30∘\sin 60^\circ \cos 30^\circ - \cos 60^\circ \sin 30^\circ = \sin(60^\circ - 30^\circ) = \sin 30^\circ

  3. Substitute the value of sin⁡30∘\sin 30^\circ.

    sin⁡30∘=12\sin 30^\circ = \frac{1}{2}

    The value is 12\boxed{\frac{1}{2}}.

Part 1.5· 2 steps

sin⁡245∘+cos⁡245∘\sin^2 45^\circ + \cos^2 45^\circ

  1. Recognize the Pythagorean identity.

    sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1

  2. Apply the identity for θ=45∘\theta = 45^\circ.

    sin⁡245∘+cos⁡245∘=1\sin^2 45^\circ + \cos^2 45^\circ = 1

    The value is 1\boxed{1}.

Part 1.6· 3 steps

1−sin⁡230∘\sqrt{1 - \sin^2 30^\circ}

  1. Recognize the Pythagorean identity 1−sin⁡2θ=cos⁡2θ1 - \sin^2 \theta = \cos^2 \theta.

    1−sin⁡230∘=cos⁡230∘\sqrt{1 - \sin^2 30^\circ} = \sqrt{\cos^2 30^\circ}

  2. Simplify the square root.

    Since 30∘30^\circ is in the first quadrant, cos⁡30∘\cos 30^\circ is positive.

    cos⁡230∘=cos⁡30∘\sqrt{\cos^2 30^\circ} = \cos 30^\circ

  3. Substitute the value of cos⁡30∘\cos 30^\circ.

    cos⁡30∘=32\cos 30^\circ = \frac{\sqrt{3}}{2}

    The value is 32\boxed{\frac{\sqrt{3}}{2}}.

Part 1.7· 3 steps

cos⁡60∘tan⁡45∘sin⁡30∘sec⁡30∘\frac{\cos 60^\circ \tan 45^\circ \sin 30^\circ}{\sec 30^\circ}

  1. Substitute the known values for each trigonometric function.

    cos⁡60∘=12\cos 60^\circ = \frac{1}{2} tan⁡45∘=1\tan 45^\circ = 1 sin⁡30∘=12\sin 30^\circ = \frac{1}{2} sec⁡30∘=1cos⁡30∘=132=23\sec 30^\circ = \frac{1}{\cos 30^\circ} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}

  2. Calculate the numerator.

    Numerator=12×1×12=14\text{Numerator} = \frac{1}{2} \times 1 \times \frac{1}{2} = \frac{1}{4}

  3. Divide the numerator by the denominator.

    1423=14×32=38\frac{\frac{1}{4}}{\frac{2}{\sqrt{3}}} = \frac{1}{4} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{8}

    The value is 38\boxed{\frac{\sqrt{3}}{8}}.

Part 1.8· 3 steps

2cos⁡245∘−12\cos^2 45^\circ - 1

  1. Recognize the double angle identity for cosine.

    cos⁡(2θ)=2cos⁡2θ−1\cos(2\theta) = 2\cos^2 \theta - 1

  2. Apply the identity with θ=45∘\theta = 45^\circ.

    2cos⁡245∘−1=cos⁡(2×45∘)=cos⁡90∘2\cos^2 45^\circ - 1 = \cos(2 \times 45^\circ) = \cos 90^\circ

  3. Substitute the value of cos⁡90∘\cos 90^\circ.

    cos⁡90∘=0\cos 90^\circ = 0

    The value is 0\boxed{0}.

Part 1.9· 3 steps

sin⁡60∘cos⁡60∘\frac{\sin 60^\circ}{\cos 60^\circ}

  1. Recognize the definition of tan⁡θ\tan \theta.

    tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}

  2. Apply the definition for θ=60∘\theta = 60^\circ.

    sin⁡60∘cos⁡60∘=tan⁡60∘\frac{\sin 60^\circ}{\cos 60^\circ} = \tan 60^\circ

  3. Substitute the value of tan⁡60∘\tan 60^\circ.

    tan⁡60∘=3\tan 60^\circ = \sqrt{3}

    The value is 3\boxed{\sqrt{3}}.

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