1 Simplify the following equation without using a calculator: 2 sec 300° – tan²(-135°) – 2 cos² 90° + 3 sin 210°
|Mathematics
1 Simplify the following equation without using a calculator: 2 sec 300° – tan²(-135°) – 2 cos² 90° + 3 sin 210°
ScanSolve AI Answer
Answer
23
Here are the solutions to the trigonometry problems.
2.1 Simplify the following equation without using a calculator:2sec300∘−tan2(−135∘)−2cos290∘+3sin210∘
Step 1: Evaluate each trigonometric term.
• sec300∘=cos300∘1=cos(360∘−60∘)1=cos60∘1=1/21=2.
• tan(−135∘)=tan(−135∘+360∘)=tan(225∘)=tan(180∘+45∘)=tan45∘=1.
So, tan2(−135∘)=(1)2=1.
• cos90∘=0.
So, 2cos290∘=2(0)2=0.
• sin210∘=sin(180∘+30∘)=−sin30∘=−21.
Step 2: Substitute these values back into the expression.
2(2)−1−0+3(−21)4−1−0−233−2326−2323
The simplified value is 23.
2.2 Show that sin(x+45∘)=2sinx+cosx
Step 1: Use the angle addition formula for sine: sin(A+B)=sinAcosB+cosAsinB.
Let A=x and B=45∘.
sin(x+45∘)=sinxcos45∘+cosxsin45∘
Step 2: Substitute the known values for cos45∘ and sin45∘.
We know that cos45∘=22 and sin45∘=22.
sin(x+45∘)=sinx(22)+cosx(22)
Step 3: Factor out 22 and simplify.
sin(x+45∘)=22(sinx+cosx)
Since 22=21, we can write:
sin(x+45∘)=21(sinx+cosx)sin(x+45∘)=2sinx+cosx
This matches the right-hand side, thus the identity is proven.
2.3 Determine the value of x∈[0∘,90∘] without using a calculator:cot(x+15∘)=tan(x−60∘)
Step 1: Use the co-function identity cotθ=tan(90∘−θ).
tan(90∘−(x+15∘))=tan(x−60∘)tan(90∘−x−15∘)=tan(x−60∘)tan(75∘−x)=tan(x−60∘)
Step 2: Solve the equation for x using the general solution for tanA=tanB, which is A=B+n⋅180∘ for integer n.
75∘−x=x−60∘+n⋅180∘75∘+60∘=x+x+n⋅180∘135∘=2x+n⋅180∘2x=135∘−n⋅180∘x=2135∘−n⋅2180∘x=67.5∘−n⋅90∘
Step 3: Find the value of x in the given range [0∘,90∘].
• For n=0: x=67.5∘−0⋅90∘=67.5∘. This value is within the range [0∘,90∘].
• For n=1: x=67.5∘−1⋅90∘=−22.5∘. This value is not in the range.
• For n=−1: x=67.5∘−(−1)⋅90∘=67.5∘+90∘=157.5∘. This value is not in the range.
The only value of x in the specified range is 67.5∘.
Step 1: Simplify the terms in the denominator using reduction formulas.
• cos(2π−x)=sinx (co-function identity).
• sin(π−y)=siny (sine is positive in the second quadrant).
Substitute these into the denominator:
Denominator=cos(x+y)+sinxsiny
Step 2: Expand cos(x+y) in the denominator.
The formula for cos(x+y) is cosxcosy−sinxsiny.
Substitute this into the denominator:
Denominator=(cosxcosy−sinxsiny)+sinxsinyDenominator=cosxcosy
Step 3: Rewrite the right-hand side (RHS) of the equation with the simplified denominator.
RHS=cosxcosysin(x+y)
Step 4: Expand sin(x+y) in the numerator.
The formula for sin(x+y) is sinxcosy+cosxsiny.
Substitute this into the RHS:
RHS=cosxcosysinxcosy+cosxsiny
Step 5: Separate the terms in the numerator and simplify.
RHS=cosxcosysinxcosy+cosxcosycosxsinyRHS=cosxsinx+cosysinyRHS=tanx+tany
This matches the left-hand side (LHS) of the equation, thus the identity is proven.
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Here are the solutions to the trigonometry problems. 2.1 Simplify the following equation without using a calculator: 2 300^ - ^2(-135^) - 2 ^2 90^ + 3 210^ Step 1: Evaluate each trigonometric term. • 300^ = (1)/( 300^) = (1)/((360^ - 60^)) = (1)/( 60^) = (1)/(1/2) = 2. • (-135^) = (-135^ + 360^) = (225^) = (180^ + 45^) = 45^ = 1. So, ^2(-135^) = (1)^2 = 1. • 90^ = 0. So, 2 ^2 90^ = 2(0)^2 = 0. • 210^ = (180^ + 30^) = - 30^ = -(1)/(2). Step 2: Substitute these values back into the expression. 2(2) - 1 - 0 + 3(-(1)/(2)) 4 - 1 - 0 - (3)/(2) 3 - (3)/(2) (6)/(2) - (3)/(2) (3)/(2) The simplified value is (3)/(2). 2.2 Show that (x + 45^) = ( x + x)/(sqrt(2)) Step 1: Use the angle addition formula for sine: (A+B) = A B + A B. Let A=x and B=45^. (x + 45^) = x 45^ + x 45^ Step 2: Substitute the known values for 45^ and 45^. We know that 45^ = sqrt(2)2 and 45^ = sqrt(2)2. (x + 45^) = x (sqrt(2)2) + x (sqrt(2)2) Step 3: Factor out sqrt(2)2 and simplify. (x + 45^) = sqrt(2)2 ( x + x) Since sqrt(2)2 = (1)/(sqrt(2)), we can write: (x + 45^) = (1)/(sqrt(2)) ( x + x) (x + 45^) = ( x + x)/(sqrt(2)) This matches the right-hand side, thus the identity is proven. 2.3 Determine the value of x [0^, 90^] without using a calculator: (x + 15^) = (x - 60^) Step 1: Use the co-function identity = (90^ - ). (90^ - (x + 15^)) = (x - 60^) (90^ - x - 15^) = (x - 60^) (75^ - x) = (x - 60^) Step 2: Solve the equation for x using the general solution for A = B, which is A = B + n · 180^ for integer n. 75^ - x = x - 60^ + n · 180^ 75^ + 60^ = x + x + n · 180^ 135^ = 2x + n · 180^ 2x = 135^ - n · 180^ x = (135^)/(2) - n · (180^)/(2) x = 67.5^ - n · 90^ Step 3: Find the value of x in the given range [0^, 90^]. • For n=0: x = 67.5^ - 0 · 90^ = 67.5^. This value is within the range [0^, 90^]. • For n=1: x = 67.5^ - 1 · 90^ = -22.5^. This value is not in the range. • For n=-1: x = 67.5^ - (-1) · 90^ = 67.5^ + 90^ = 157.5^. This value is not in the range. The only value of x in the specified range is 67.5^. 2.4 Prove that: x + y = ((x+y))/((x+y)+()2-x)(-y) Step 1: Simplify the terms in the denominator using reduction formulas. • (()/(2)-x) = x (co-function identity). • (-y) = y (sine is positive in the second quadrant). Substitute these into the denominator: Denominator = (x+y) + x y Step 2: Expand (x+y) in the denominator. The formula for (x+y) is x y - x y. Substitute this into the denominator: Denominator = ( x y - x y) + x y Denominator = x y Step 3: Rewrite the right-hand side (RHS) of the equation with the simplified denominator. RHS = ((x+y))/( x y) Step 4: Expand (x+y) in the numerator. The formula for (x+y) is x y + x y. Substitute this into the RHS: RHS = ( x y + x y)/( x y) Step 5: Separate the terms in the numerator and simplify. RHS = ( x y)/( x y) + ( x y)/( x y) RHS = ( x)/( x) + ( y)/( y) RHS = x + y This matches the left-hand side (LHS) of the equation, thus the identity is proven. 4 down. 1 left today.
