Trigonometry Quiz: Angle Addition And Subtraction

Explanation
The correct answer is (√2 + √6)/4. This can be determined by using the cosine function and the unit circle. The cosine of 105 degrees can be found by evaluating the x-coordinate of the point on the unit circle corresponding to 105 degrees. By using the values of √2 and √6, which are the x-coordinates of the points on the unit circle corresponding to 45 degrees and 60 degrees respectively, we can add them together and divide by 4 to get the cosine of 105 degrees.

Explanation
In the given question, it is stated that sin B = -5/13. Since the range of A is π

Explanation
The given expression, SinA.CosB - CosA.SinB, can be simplified using the trigonometric identity for the sine of the difference of two angles, which is Sin (A - B). Therefore, the correct answer is Sin (A - B).

Explanation
The given information states that tan(A + B) = √3 and tan(A – B) = 1/√3. Since tan(A + B) = √3, we can conclude that the angle (A + B) must be 60 degrees, as the tangent of 60 degrees is √3. Similarly, tan(A – B) = 1/√3 implies that the angle (A – B) must be 30 degrees, as the tangent of 30 degrees is 1/√3.

Now, we can solve for A by adding the two equations: (A + B) + (A – B) = 60 + 30. This simplifies to 2A = 90, and dividing both sides by 2 gives A = 45 degrees. Therefore, the correct answer is 45 degrees.

Explanation
The given information states that tan(A + B) = √3 and tan(A – B) = 1/√3. Since the values of tan(A + B) and tan(A – B) are known, we can use the trigonometric identities to find the value of B. By using the identity tan(A + B) = (tanA + tanB) / (1 - tanA * tanB), we can substitute the given values to get (√3) = (tanA + tanB) / (1 - tanA * tanB). Similarly, using the identity tan(A – B) = (tanA - tanB) / (1 + tanA * tanB), we can substitute the given values to get (1/√3) = (tanA - tanB) / (1 + tanA * tanB). By solving these two equations simultaneously, we can find the value of B, which is 15 degrees.

Explanation
The given information states that cos A is equal to -4/5, where A is an angle between π/2 and π. Since cosine is equal to the adjacent side divided by the hypotenuse in a right triangle, we can conclude that the adjacent side is -4 and the hypotenuse is 5. By using the Pythagorean theorem, we can find the opposite side, which is 3. Therefore, the sine of A is equal to 3/5.

Explanation
The given equation sin 3A = cos (A – 26°) implies that the sine of 3A is equal to the cosine of (A – 26°). Since the sine and cosine functions have a phase difference of 90°, we can conclude that 3A and (A – 26°) must also have a phase difference of 90°. This means that 3A and (A – 26°) are complementary angles. Since A is an acute angle, the only possible value for A that satisfies this condition is 29 degrees.

Explanation
To solve the given equation (√6 - √2)/4, we can simplify it by rationalizing the denominator. We multiply both the numerator and the denominator by the conjugate of the denominator, which is (√6 + √2). This results in (√6 - √2)(√6 + √2) = (√6)^2 - (√2)^2 = 6 - 2 = 4. Therefore, the simplified equation becomes 4/4, which simplifies further to 1. The value of sin 15 degrees is also 1/4. Hence, the answer is sin 15 degrees.

Explanation
The given expression, CosA.CosB - SinA.SinB, is the correct answer because it represents the formula for the cosine of the sum of two angles. The formula states that the cosine of the sum of angles A and B is equal to the product of the cosines of each angle minus the product of the sines of each angle. Therefore, the correct answer is CosA.CosB - SinA.SinB.

Explanation
The given information states that A is an angle between π/2 and π, and that Tan A is equal to -4/5. The only option that satisfies this condition is -3/4.