The question will first read in English followed by a reading in French.
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If alpha + 2 beta = pi/4 AND tan beta = 1/3. What would tan alpha mesure?
(Keep in mind that pi/4 is the same as 45 degrees.)
Si alpha + 2 beta = pi/4 ET tan beta = 1/3. Calculer exactement tan alpha. (Gardez en tête que pi/4 est égal à 45 degrés.)
To find the value of tan alpha, we need to manipulate the equation alpha + 2 beta = pi/4 and use the given information tan beta = 1/3.
1. Start with the equation alpha + 2 beta = pi/4.
2. Let's solve for alpha by subtracting 2 beta from both sides of the equation:
alpha = pi/4 - 2 beta.
3. Now we have an expression for alpha in terms of beta.
4. We also know that tan beta = 1/3.
5. Recall that tan is the ratio of the sine of an angle to the cosine of the angle:
tan = sin / cos.
6. We can substitute the values of tan beta into the equation:
1/3 = sin(beta) / cos(beta).
7. Rearrange the equation to solve for sin(beta):
sin(beta) = 1/3 * cos(beta).
8. Now we can use the identity sin^2(beta) + cos^2(beta) = 1 to find cos(beta).
9. Substitute sin^2(beta) = (1/3)^2 * cos^2(beta) into the identity:
(1/3)^2 * cos^2(beta) + cos^2(beta) = 1.
10. Simplify the equation:
1/9 * cos^2(beta) + cos^2(beta) = 1.
11. Combine the terms:
(1/9 + 1) * cos^2(beta) = 1.
12. Find the common denominator and add the fractions:
(10/9) * cos^2(beta) = 1.
13. Solve for cos^2(beta):
cos^2(beta) = 9/10.
14. Taking the square root of both sides, we get:
cos(beta) = sqrt(9/10) = 3/√10.
15. Since we know cos(beta), we can find sin(beta) using the identity sin^2(beta) + cos^2(beta) = 1:
sin^2(beta) = 1 - (3/√10)^2 = 1 - 9/10 = 1/10.
16. Taking the square root of both sides, we get:
sin(beta) = sqrt(1/10) = 1/√10.
17. Finally, we can substitute the values of sin(beta) and cos(beta) into the equation for tan alpha:
tan(alpha) = sin(alpha) / cos(alpha).
18. Using the values of sin(beta) and cos(beta), we get:
tan(alpha) = (1/√10) / (3/√10) = 1/3.
Therefore, tan alpha measures 1/3.
To find the value of tan alpha, we'll start with the given equations and use trigonometric identities to solve for tan alpha.
Given:
1. alpha + 2 beta = pi/4
2. tan beta = 1/3
Let's solve this step by step:
Step 1: Solve equation 1 for alpha.
alpha = pi/4 - 2 beta
Step 2: Substitute the value of alpha from step 1 into the equation for tan alpha.
tan alpha = tan(pi/4 - 2 beta)
Step 3: Use the tangent difference identity: tan(a - b) = (tan a - tan b) / (1 + tan a tan b).
tan alpha = (tan(pi/4) - tan(2 beta)) / (1 + tan(pi/4) tan(2 beta))
Step 4: Simplify the expression.
tan alpha = (1 - tan(2 beta)) / (1 + tan(2 beta))
Step 5: Substitute the value of tan beta from the given equation in step 2.
tan alpha = (1 - tan(2 beta)) / (1 + (1/3) tan beta)
Step 6: Evaluate the expression using the given values.
tan alpha = (1 - tan(2 beta)) / (1 + (1/3) * (1/3))
= (1 - tan(2 beta)) / (1 + 1/9)
= (1 - tan(2 beta)) / (10/9)
= 9(1 - tan(2 beta)) / 10
Now, we need to calculate tan(2 beta).
Given:
tan beta = 1/3
Step 7: Use the tangent double angle identity: tan(2 beta) = 2 tan beta / (1 - tan^2 beta).
tan(2 beta) = 2 * (1/3) / (1 - (1/3)^2)
= 2/3 / (1 - 1/9)
= 2/3 / (8/9)
= 2/3 * 9/8
= 3/4
Step 8: Substitute the value of tan(2 beta) from step 7 into the expression derived in step 6.
tan alpha = 9(1 - 3/4) / 10
= 9/10 - (27/40)
= (9 - 27/4) / 10
= (36 - 27)/40
= 9/40
Therefore, tan alpha measures 9/40.