Find the exact value of each of the following under the given conditions below.
[tex]$\tan \alpha=-\frac{3}{4}, \frac{\pi}{2}\ \textless \ \alpha\ \textless \ \pi ; \sin \beta=\frac{1}{2}, 0\ \textless \ \beta\ \textless \ \frac{\pi}{2}[/tex]
(a) [tex]$\sin (\alpha+\beta)$[/tex]
(b) [tex]$\cos (\alpha+\beta)$[/tex]
(c) [tex]$\sin (\alpha-\beta)$[/tex]
(d) [tex]$\tan (\alpha-\beta)$[/tex]
Answer (2)
To find the trigonometric values, we first obtained sin α and cos α using the given tan α . Then, we calculated sin β and cos β for the angle values and used trigonometric identities to derive each requested value. The exact answers for sin ( α + β ) , cos ( α + β ) , sin ( α − β ) , and tan ( α − β ) were found to be 10 3 3 − 4 , 10 − 4 3 − 3 , 10 3 3 + 4 , and − 39 48 + 25 3 respectively.
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Find sin α and cos α using the given tan α and the quadrant information: sin α = 5 3 , cos α = − 5 4 .
Find cos β using the given sin β and the quadrant information: cos β = 2 3 .
Apply the angle sum and difference formulas to find sin ( α + β ) , cos ( α + β ) , sin ( α − β ) , and tan ( α − β ) .
The final answers are: sin ( α + β ) = 10 3 3 − 4 , cos ( α + β ) = 10 − 4 3 − 3 , sin ( α − β ) = 10 3 3 + 4 , tan ( α − β ) = − 39 48 + 25 3 .
Explanation
Analyze the problem and given data We are given that tan α = − 4 3 and 2 π < α < π . This means α is in the second quadrant, where sine is positive and cosine is negative. We are also given that sin β = 2 1 and 0 < β < 2 π , which means β is in the first quadrant, where both sine and cosine are positive. Our goal is to find the exact values of sin ( α + β ) , cos ( α + β ) , sin ( α − β ) , and tan ( α − β ) .
Find sin α and cos α First, we need to find sin α and cos α . Since tan α = − 4 3 , we can think of a right triangle with opposite side 3 and adjacent side 4. The hypotenuse is 3 2 + 4 2 = 9 + 16 = 25 = 5 . Since α is in the second quadrant, sin α = 5 3 and cos α = − 5 4 .
Find cos β Next, we need to find cos β . Since sin β = 2 1 and β is in the first quadrant, we can use the Pythagorean identity sin 2 β + cos 2 β = 1 to find cos β . Thus, cos 2 β = 1 − sin 2 β = 1 − ( 2 1 ) 2 = 1 − 4 1 = 4 3 . Since β is in the first quadrant, cos β = 4 3 = 2 3 .
Calculate sin ( α + β ) (a) Now we can find sin ( α + β ) using the angle sum formula: sin ( α + β ) = sin α cos β + cos α sin β = ( 5 3 ) ( 2 3 ) + ( − 5 4 ) ( 2 1 ) = 10 3 3 − 10 4 = 10 3 3 − 4 ≈ 0.1196 .
Calculate cos ( α + β ) (b) Next, we find cos ( α + β ) using the angle sum formula: cos ( α + β ) = cos α cos β − sin α sin β = ( − 5 4 ) ( 2 3 ) − ( 5 3 ) ( 2 1 ) = − 10 4 3 − 10 3 = 10 − 4 3 − 3 ≈ − 0.9928 .
Calculate sin ( α − β ) (c) Now we find sin ( α − β ) using the angle difference formula: sin ( α − β ) = sin α cos β − cos α sin β = ( 5 3 ) ( 2 3 ) − ( − 5 4 ) ( 2 1 ) = 10 3 3 + 10 4 = 10 3 3 + 4 ≈ 0.9196 .
Calculate tan ( α − β ) (d) Finally, we find tan ( α − β ) . First, we need to find tan β = c o s β s i n β = 2 3 2 1 = 3 1 = 3 3 . Then, we use the angle difference formula for tangent: tan ( α − β ) = 1 + t a n α t a n β t a n α − t a n β = 1 + ( − 4 3 ) ( 3 3 ) − 4 3 − 3 3 = 1 − 4 3 − 4 3 − 3 3 = 4 4 − 3 − 12 9 + 4 3 = 3 ( 4 − 3 ) − ( 9 + 4 3 ) = 3 ( 4 − 3 ) ( 4 + 3 ) − ( 9 + 4 3 ) ( 4 + 3 ) = 3 ( 16 − 3 ) − ( 36 + 9 3 + 16 3 + 12 ) = 39 − ( 48 + 25 3 ) = − 39 48 + 25 3 ≈ − 2.3411 .
State the final answer Therefore, the exact values are: (a) sin ( α + β ) = 10 3 3 − 4 (b) cos ( α + β ) = 10 − 4 3 − 3 (c) sin ( α − β ) = 10 3 3 + 4 (d) tan ( α − β ) = − 39 48 + 25 3
Examples
Understanding trigonometric identities and angle sum/difference formulas is crucial in various fields such as physics and engineering. For example, when analyzing the motion of a projectile, we often need to decompose the initial velocity into horizontal and vertical components using trigonometric functions. Similarly, in electrical engineering, alternating current (AC) circuits involve sinusoidal functions, and understanding phase shifts and combinations of these functions requires a solid grasp of trigonometric identities.
