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
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°
ScanSolve AI Answer
Answer
323
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Here are the solutions for the trigonometric expressions:
1.1 sec30∘
Step 1: Recall the definition of secθ.
secθ=cosθ1
Step 2: Substitute the value of cos30∘.
cos30∘=23
Step 3: Calculate sec30∘.
sec30∘=231=32
Step 4: Rationalize the denominator.
sec30∘=32×33=323The value is323.
1.2 cos60∘+tan45∘
Step 1: Substitute the known values.
cos60∘=21tan45∘=1
Step 2: Add the values.
cos60∘+tan45∘=21+1=21+22=23The value is23.
1.3 cos30∘×sin60∘
Step 1: Substitute the known values.
cos30∘=23sin60∘=23
Step 2: Multiply the values.
cos30∘×sin60∘=23×23=2×23×3=43The value is43.
1.4 sin60∘cos30∘−cos60∘sin30∘
Step 1: Recognize the trigonometric identity for sin(A−B).
sin(A−B)=sinAcosB−cosAsinB
Step 2: Apply the identity with A=60∘ and B=30∘.
sin60∘cos30∘−cos60∘sin30∘=sin(60∘−30∘)=sin30∘
Step 3: Substitute the value of sin30∘.
sin30∘=21The value is21.
1.5 sin245∘+cos245∘
Step 1: Recognize the Pythagorean identity.
sin2θ+cos2θ=1
Step 2: Apply the identity for θ=45∘.
sin245∘+cos245∘=1The value is1.
1.6 1−sin230∘
Step 1: Recognize the Pythagorean identity 1−sin2θ=cos2θ.
1−sin230∘=cos230∘
Step 2: Simplify the square root. Since 30∘ is in the first quadrant, cos30∘ is positive.
cos230∘=cos30∘
Step 3: Substitute the value of cos30∘.
cos30∘=23The value is23.
1.7 sec30∘cos60∘tan45∘sin30∘
Step 1: Substitute the known values for each trigonometric function.
cos60∘=21tan45∘=1sin30∘=21sec30∘=cos30∘1=231=32
Step 2: Calculate the numerator.
Numerator=21×1×21=41
Step 3: Divide the numerator by the denominator.
3241=41×23=83The value is83.
1.8 2cos245∘−1
Step 1: Recognize the double angle identity for cosine.
cos(2θ)=2cos2θ−1
Step 2: Apply the identity with θ=45∘.
2cos245∘−1=cos(2×45∘)=cos90∘
Step 3: Substitute the value of cos90∘.
cos90∘=0The value is0.
1.9 cos60∘sin60∘
Step 1: Recognize the definition of tanθ.
tanθ=cosθsinθ
Step 2: Apply the definition for θ=60∘.
cos60∘sin60∘=tan60∘
Step 3: Substitute the value of tan60∘.
tan60∘=3The value is3.
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This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
You're on a roll — Here are the solutions for the trigonometric expressions: 1.1 30^ Step 1: Recall the definition of . = (1)/( ) Step 2: Substitute the value of 30^. 30^ = sqrt(3)2 Step 3: Calculate 30^. 30^ = (1)/(sqrt(3))2 = (2)/(sqrt(3)) Step 4: Rationalize the denominator. 30^ = (2)/(sqrt(3)) × sqrt(3)sqrt(3) = 2sqrt(3)3 The value is 2sqrt(3)3. 1.2 60^ + 45^ Step 1: Substitute the known values. 60^ = (1)/(2) 45^ = 1 Step 2: Add the values. 60^ + 45^ = (1)/(2) + 1 = (1)/(2) + (2)/(2) = (3)/(2) The value is (3)/(2). 1.3 30^ × 60^ Step 1: Substitute the known values. 30^ = sqrt(3)2 60^ = sqrt(3)2 Step 2: Multiply the values. 30^ × 60^ = sqrt(3)2 × sqrt(3)2 = sqrt(3) × sqrt(3)2 × 2 = (3)/(4) The value is (3)/(4). 1.4 60^ 30^ - 60^ 30^ Step 1: Recognize the trigonometric identity for (A-B). (A-B) = A B - A B Step 2: Apply the identity with A=60^ and B=30^. 60^ 30^ - 60^ 30^ = (60^ - 30^) = 30^ Step 3: Substitute the value of 30^. 30^ = (1)/(2) The value is (1)/(2). 1.5 ^2 45^ + ^2 45^ Step 1: Recognize the Pythagorean identity. ^2 + ^2 = 1 Step 2: Apply the identity for = 45^. ^2 45^ + ^2 45^ = 1 The value is 1. 1.6 sqrt(1 - ^2 30^) Step 1: Recognize the Pythagorean identity 1 - ^2 = ^2 . sqrt(1 - ^2 30^) = sqrt(^2 30^) Step 2: Simplify the square root. Since 30^ is in the first quadrant, 30^ is positive. sqrt(^2 30^) = 30^ Step 3: Substitute the value of 30^. 30^ = sqrt(3)2 The value is sqrt(3)2. 1.7 ( 60^ 45^ 30^)/( 30^) Step 1: Substitute the known values for each trigonometric function. 60^ = (1)/(2) 45^ = 1 30^ = (1)/(2) 30^ = (1)/( 30^) = (1)/(sqrt(3))2 = (2)/(sqrt(3)) Step 2: Calculate the numerator. Numerator = (1)/(2) × 1 × (1)/(2) = (1)/(4) Step 3: Divide the numerator by the denominator. (1)/(4)(2)/(sqrt(3)) = (1)/(4) × sqrt(3)2 = sqrt(3)8 The value is sqrt(3)8. 1.8 2^2 45^ - 1 Step 1: Recognize the double angle identity for cosine. (2) = 2^2 - 1 Step 2: Apply the identity with = 45^. 2^2 45^ - 1 = (2 × 45^) = 90^ Step 3: Substitute the value of 90^. 90^ = 0 The value is 0. 1.9 ( 60^)/( 60^) Step 1: Recognize the definition of . = ( )/( ) Step 2: Apply the definition for = 60^. ( 60^)/( 60^) = 60^ Step 3: Substitute the value of 60^. 60^ = sqrt(3) The value is sqrt(3).
