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# Common Core Standard HSF-TF.C.9 Questions

(+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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What is the missing reason in step 18?
1. Reciprocal Identity
2. Radii of the same circle are equal
3. Transitive Property of Equality
4. Pythagorean Theorem

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What is the missing expression in step 5?
1.  $( ( \ sin(alpha)cos(beta) + sin(beta)cos(alpha) \ ))/( ( \ cos(alpha)cos(beta) \ ) ( \ cos(alpha)cos(beta) - sin(alpha)sin(beta) \ ))$ $\ \$
2.  $( ( \ cos(alpha)cos(beta) \ )( \ sin(alpha)cos(beta) + sin(beta)cos(alpha) \ ))/(( \ cos(alpha)cos(beta) - sin(alpha)sin(beta) \ ))$ $\ \$
3.  $( ( \ cos(alpha)cos(beta) \ )( \ sin(alpha)cos(beta) + sin(beta)cos(alpha) \ ))/( ( \ cos(alpha)cos(beta) \ ) ( \ cos(alpha)cos(beta) - sin(alpha)sin(beta) \ ))$ $\ \$
4.  $( ( \ sin(alpha)cos(beta) \ )( \ sin(beta)cos(alpha) \ ))/( ( \ cos(alpha)cos(beta) \ ) ( \ sin(alpha)sin(beta) \ ))$ $\ \$

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Which of the following is equal to $cos(-theta) ?$
1. $cos(theta)$
2. $-cos(theta)$
3. $-cos(-theta)$
4. $cos(theta-1)$

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What is the missing explanation in step 1?
1. $"Definition of tangent"$
2. $"Pythagorean Identity"$
3. $"Law of S""ines"$
4. $alpha/alpha = 1, \ beta/beta = 1$

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Which of the following is equal to $sin(-theta) ?$
1. $sin(theta)$
2. $-sin(theta)$
3. $-sin(-theta)$
4. $sin(theta-1)$

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What is the missing expression in step 2?
1. $(((sin(alpha)cos(beta))/(cos(alpha)sin(beta))))/((1-sin(alpha)/cos(alpha) sin(beta)/cos(beta)))$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$
2. $(((sin(alpha)sin(beta))/(cos(alpha)cos(beta))))/((1-sin(alpha)/cos(alpha) sin(beta)/cos(beta)))$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$
3. $(((sin(alpha)cos(alpha) \ + \ sin(beta)cos(beta))/(cos(alpha)cos(beta))))/((1-sin(alpha)/cos(alpha) sin(beta)/cos(beta)))$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$
4. $(((sin(alpha)cos(beta) \ + \ sin(beta)cos(alpha))/(cos(alpha)cos(beta))))/((1-sin(alpha)/cos(alpha) sin(beta)/cos(beta)))$

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What is the missing explanation in step 3?
1. $"The sine and cosine functions are never greater than 1"$
2. $cos^2(theta)+ sin^2(theta)=1$
3. $cos(0)=1, \ sin(90°)=1$
4. $"The number 1 can be rewritten as a given quantity divided by itself"$

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What is the missing reason in step 4?
1. Distributive Property of Multiplication
2. Commutative Property of Multiplication
3. SSA formula for area of a triangle
4. Law of Sines

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What is the missing explanation in step 6?
1. Pythagorean Identity
2. Cancel like factors
3. Substitution Property of Equality
4. Division Property of Identity

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What are some of the implicit restrictions in the above proof? Choose all correct answers.
1. $0 < theta+alpha < pi$, where $theta,alpha$ must both be positive real numbers.
2. $0 < theta+alpha < pi/2$, where $theta, alpha$ must both be positive real numbers.
3. $Delta ABC$ cannot be a right triangle.
4. $ang BAC$ and $angBCA$ must both be acute angles.

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What is the missing reason in step 7?
1. Given
2. Right triangles have 2 sets of perpendicular sides
3. Triangle Property
4. SSS Property of Similarity

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What is the missing explanation in step 7?
1. Substitute in the sine addition formula
2. Apply the Law of Sines
3. Cancel like terms
4. Factor out like terms

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What is the missing statement in step 9?
1. $ang ABC$ is a right angle
2. Point $D$ is the midpoint of $bar{AC}$
3. $ang ABC$ is acute
4. $ang ADB, \ ang BDC$ are right angles