Test: Trigonometry- 1 - GMAT MCQ

Step 1: Question statement and Inferences

Let us represent the two cases diagrammatically.

Left Triangle BAC represents the initial case. Point A is on the line of intersection of the wall and the floor.

We are given that AB = AC and that BC = 60 feet.

Right Triangle PAQ represents the case after the ladder is moved. The only value that remains unchanged after the movement is the ladder’s length. Hence, in Right Triangle PAQ,

PQ = 60 feet, and Angle AQP = 30°

We need to find the value of AB – AP.

Step 2: Finding required values

Let us first consider Right Triangle BAC:

We know that when two sides of a right triangle are equal, it is a 45°-45°-90° triangle.

In such a triangle, the sides opposite to the angles 45°, 45°, and 90° respectively are in the ratio 1 : 1 : √2 

Therefore,

Or,

 Thus, we have found out the value of AB.

Now, considering Right Triangle PAQ:

It is a 30°-60°-90° right angled triangle. 

We know that in such a triangle, the sides opposite to the angles 30°, 60°, and 90° respectively are in  the ratio 1 : √3 : 2.

Therefore,

or,

Step 3: Calculate the final answer

Difference between the heights to which the ladder reaches in the two cases

Answer: Option (B)

We are given that                               AngleABC=30∘

△ABC   is a right-triangle (as indicated in the given figure)

Therefore, from the Angle Sum Property:

 Angle ABC+ Angle BCA+ Angle CAB = 180^∘                       

30^∘ +90^∘+ AngleCAB =180^∘

Thus, 

                                                          AngleCAB=60^∘

Therefore, △ABC  is a 30°-60°-90° right angled triangle. In other words, the ladder forms a 30°-60°-90° right angled triangle with the wall.

We know that in such a triangle, the sides opposite to the angles 30°, 60°, and 90° respectively are in the ratio 1 : √3 : 2.

Thus, z : y : x = 1 : √3 : 2

This implies, y = z√3

And, x = 2z

So, the sum x + y + z = 2z + z√3 + z

x + y + z = z(3+√3)

Thus, in order to find the value of the sum, we need to know the value of z or of any other side of the triangle ABC.

Step 3: Analyze Statement 1

The ratio of sides AB:BC:AC is 2: √3 : 1

We are given the ratio of sides of triangle. We have already deduced this ratio of sides in Steps 1 and 2. So Statement 1 gives us no new information.

This question illustrates well the importance of first extracting the maximum juice from the question statement before moving to Statement 1. A student who skips doing this and straightaway jumps to Statement 1 runs the risk of choosing (C) as the answer in this question.

Statement 1 is clearly Not Sufficient.

Step 4: Analyze Statement 2

The length of the ladder is 12 feet.

This means, x = 12 feet

We also know that x = 2z and y = z√3

Hence, we can easily find out the values of y and z.

Thus Statement 2 alone is Sufficient.

Step 5: Analyze Both Statements Together (if needed)

We get a unique answer in step 4, so this step is not required

Answer: Option (B)

Steps 1 & 2: Understand Question and Draw Inferences

The flexibility of the crane’s arm is an important point here. Remember that the ninth floor is at a fixed height. Hence when the crane moves closer to the building, its elevation angle will increase for sure and its arm length will decrease. Hence the only constant here is the height of the ninth floor (from the ground).

Hence, in order to find out the distance moved by the crane, we first need to find the constant height.

Let’s represent the given information diagrammatically.

In right triangle ABC, AB represents the building till the ninth floor. The length of AB is assumed to be h feet. BC represents the arm of the crane initially, when the angle of elevation was 30. From the question statement, we know that BC = 180 feet.

Since the crane’s height is given to be negligible, we have shown the arm of the crane as starting from the ground level only. Thus, points C and A lie on the horizontal ground, and the line AC is therefore perpendicular to the building AB.

Right triangle BAC is a 30°-60°-90° triangle.

We know that in such a triangle, the sides opposite to the angles 30°, 60°, and 90° respectively are in  the ratio 1 : √3 : 2.

So, h = 90 feet.

Also,          

Let the crane move x feet towards the building to point D.

Let the distance AD = y feet.

Thus, x + y = 90√3

So, if we are able to find the value of y, we will be able to find the value of x.

We have only one piece of information about right triangle DAB, that AB = 90 feet.

If we get one more piece of information about this triangle, we will be able to find the value of y.

Thus, in order to find the value of y, we need to know either one of the two unknown sides or one of the two unknown angles of Triangle DAB.

Step 3: Analyze Statement 1

After the movement, the elevation angle of the crane’s arm is 60

This means that Triangle DAB is a 30°-60°-90° triangle.

We know that in such a triangle, the sides opposite to the angles 30°, 60°, and 90° respectively are in  the ratio 1 : √3 : 2.

Thus, y =  30√3

Since we know the value of y, we will be able to find the value of x.

Thus, Statement 1 alone is sufficient to answer the question.

(Note: We have shown the full calculation here just to demonstrate how the value of y may be calculated, if a PS question was made on the same situation. Since this is a Data Sufficiency question, you need not actually solve till the value of y. Once you analyze that the given data is sufficient to find the value of y, you can move on to Step 4)

Step 4: Analyze Statement 2

After the movement, the crane is at a distance of 30√3 feet from the building.

That is, y = 30√3

Since we know the value of y, we will be able to calculate the value of x.

Thus, Statement 2 is Sufficient.

Step 5: Analyze Both Statements Together (if needed)

We get a unique answer in steps 3 and 4, so this step is not required

Answer: Option (D)

In order to make sense of the given information, let’s depict it visually:

We have assumed the lengths of sides BC, AC and AB to be a, b and c cm respectively.

CD is the perpendicular drawn from point C on the side AB.

Since CD bisects AB, AD=DB=c/2

We need to find out area of triangle ACB. We know that,

Area of a right angled triangle =1/2 x base x height

So, Area of triangle ACB =1/2 × AC × BC = 1/2 × AB × CD

     ....Equation 1

Thus, in order to calculate the area of DACB, we need to know either the values of a and b or the value of c.

Step 2: Finding required values

   In Right Triangle ACB,

  In Right Triangle ADC,

By comparing Equations 2 and 3, we get:

This simplifies to:    

c²=2b²

Or,

c=√2b                                   . . . Equation 4

In Right Triangle ACB,

By Pythagoras Theorem:

                           a²+b²=c²

Substituting Equation 4 in this equation, we get:

              a²+b²=2b²

Or, 

         a²=b²

Which means, a = b

Since two sides of right triangle ACB are equal, this means it is a 45°-45°-90° triangle.

Therefore, Angle CAB = Angle ABC = 45°

Right Triangle ADC too will therefore be a 45°-45°-90° triangle.

Therefore, in right triangle ADC, the sides opposite to the angles 45°, 45°, and 90° respectively are in the ratio 1 : 1 : √2

So, AD : CD : AC =  1 : 1 : √2       . . . Equation 5

Step 3: Calculate the final answer

From Equation 5,

Or, AD = CD = 5

c/2=5    

Thus, c=10 

From Equation 1,

Area of triangle ACB

Steps 1 & 2: Understand Question and Draw Inferences

Horizontal distance between the two posts = BC = AD

Thus, the question wants us to find the length of BC or AD.

Also, since the angle of depression from point E to point A is given to be 45°, this means that Angle EAD = 45°

Let’s represent this information in the diagram:

We know that if one of the angles of a right triangle is 45°, then the third angle would be 45° too.

Also, we know that in such a triangle, the sides opposite to the angles 45°, 45°, and 90° respectively are in the ratio 1 : 1 : √2

Thus, in right triangle ADE,

AD : ED : AE =  1 : 1 : √2  . . . Let’s call this Equation 1.

From Equation 1 it is clear that to find the length of AD, we need to know the value of ED or AE.

Step 3: Analyze Statement 1

The height of the post CE is 10 feet

Thus, DC + ED = 10 . . . Equation 2

We need to know the value of ED.

But Equation 2 is a linear equation with 2 unknowns (DC and ED). Therefore, we will not be able to find a unique value of ED.

Thus Statement 1 alone is not sufficient.

Step 4: Analyze Statement 2

The post CE is taller than the post AB by 3 feet

Thus, CE – AB =3

Or, ED=3

Putting this value of ED in Equation 1, we will be able to find the value of AD.

Thus, Statement 2 alone is sufficient to arrive at a unique answer.

Step 5: Analyze Both Statements Together (if needed)

We get a unique answer in step 4, so this step is not required

Answer: Option (B)

Steps 1 & 2: Understand Question and Draw Inferences

In this question, the angle between the horizontal and John’s line of sight when he looks at object A from point L is given to be 60^o.

This means, in right triangle LMA, Angle MAL = 60°

By using the Angle Sum Property, we can find that Angle ALM = 30°

Triangle AML is a 30° -60° -90° triangle. In such a triangle, the sides opposite to the angles 30°, 60°, and 90° respectively are in the ratio 1 : √3 : 2

That is, MA: LM : AL = 1 : √3 : 2

Now, we need to find the distance between the bottom of the lighthouse and object A. That is, we need to find the length of MA.

Let the length of MA be x feet.

Thus,LM=3√x

And, AL=2x

John then moves downstairs to the middle of the lighthouse. Let the middle point be C.

So, MC=LC=√3x/2

Let the angle of depression from Point C to object A be y°

By Applying Pythagoras Theorem in right triangle AMC, we get:

AC²=MC²+AM²

Thus, we have expressed all sides of the right triangles AML and AMC in terms of x. If we know any of these sides, we will be able to find the value of x.

Step 3: Analyze Statement 1

The distance between the top of the lighthouse and object A is 80 feet

                                                                        AL=80

This means

2x=80

x=40

Thus, Statement 1 alone is sufficient.

Step 4: Analyze Statement 2

The distance between the middle of the lighthouse and object A is 20√7 feet

 AC=20√7

This means, 

x=40

Thus, Statement 2 alone is sufficient.

Step 5: Analyze Both Statements Together (if needed)

There is no need to analyze further as both statements (1) and (2) are sufficient.

Step 1: Question statement and Inferences

The question asks us to find the length of NL.

Let NL = x feet

We are given that the angle between the horizontal and a person’s line of sight when he looks at point T from point L is 30^o.

This means, in right triangle LNT, Angle NTL =  30∘

Let’s depict this information in the figure:

Step 2: Finding required values

By applying the Angle Sum Property in right triangle LNT, we get

Angle TLN = 60°

Thus Triangle LNT is a 30°-60°-90° triangle. In a 30°-60°-90° triangle, the sides opposite to the angles 30°, 60°, and 90° respectively are in the ratio 1 : √3 : 2

That is, LN: NT : TL = 1 : √3 : 2

Also, in quadrilateral MPTN, lines MN and PT are both drawn perpendicular to the same line MP. Therefore, lines MN and PT are parallel to one another. Also, line NT is perpendicular to the vertical line MN; therefore, line NT is parallel to the horizontal line MP.

Thus, quadrilateral MPTN is made up of two sets of parallel lines and the enclosed angle is 90°

This means, quadrilateral MPTN is a rectangle.

Now, the opposite sides of a rectangle are equal.

Therefore, NT = MP = 500 feet

Step 3: Calculating the final answer

Therefore,

Hence, the correct answer is Option A

Steps 1 & 2: Understand Question and Draw Inferences

Here we can see that triangles ABC and ABD form the special triangles 30°-60°-90° and 45°-45°-90° respectively.

And we know that:

            i)      In a 30°-60°-90° triangle, the sides opposite to the angles 30°, 60°, and 90° respectively are in the ratio 1 : √3 : 2

            ii)     In a 45°-45°-90° triangle, the sides opposite to the angles 45°, 45°, and 90° respectively are in the ratio 1 : 1: √2 

Let’s assume the length of side BC to be x units.

The lengths of all the other sides of the two triangles can be calculated in terms of x by applying the above ratios. The result will be as shown:

Thus, in order to find the length of AC, we need to find the value of x.

Step 3: Analyze Statement 1

                                                           AD²+AC²=1000

This means 

  (√6x)²+(2x)²=1000

 6x²+4x²=1000

10x²=1000

x²=100

x=10

(The negative value of x is rejected because x, being the length of a geometrical figure, cannot be negative)

Since we know the value of x, we will be able to find the length of AC.

Thus, Statement 1 alone is sufficient.

Step 4: Analyze Statement 2

  CD=10(√3+1)

This means, 

 x+3√x=10(3√+1)

 x(1+√3)=10(√3+1)

x=10

Since we know the value of x, we will be able to find the length of AC.

Thus, Statement 2 alone is sufficient

Step 5: Analyze Both Statements Together (if needed)

We do not need to analyze further as both statements (1) and (2) are sufficient.

Answer: Option (D)

aSinx + bcosx can take a range of values. What is the possible range in this scenario?

5(sinx cosA + sinA cosx) ≥ -r
5(sin(x + A)) ≥ -r
5sin (x + A) ≥ -r
-1 ≤ sin (angle) ≤ 1
5sin (x + A) ≥ -5
r_min = 5
The question is "What is the smallest value ‘r’ can to take?"

Hence, the answer is 5.

Number x lies between 0 and 1, higher powers of the number will be smaller than lower powers. Think about that. And then think about what does this have to do with Trigonometry.

We know that Sin²x + Cos²x = 1 for all values of x.
If Sin x or Cos x is equal to –1 or 1, then Sin²⁰¹⁴x + Cos²⁰¹⁴x will be equal to 1.

Sin x is equal to –1 or 1 when x = –4.5π or –3.5π or –2.5π or –1.5π or –0.5π or 0.5π or 1.5π or 2.5π or 3.5π or 4.5π.

Cosx is equal to –1 or 1 when x = –5π or –4π or –3π or –2π or –π or 0 or π or 2π or 3π or 4π or 5π.

For all other values of x, Sin²⁰¹⁴ x will be strictly lesser than Sin²x.
For all other values of x, Cos²⁰¹⁴ x will be strictly lesser than Cos²x.

We know that Sin²x + Cos²x is equal to 1. Hence, Sin²⁰¹⁴x + Cos²⁰¹⁴x will never be equal to 1 for all other values of x. Thus there are 21 values.

The question is "How many values the 'x' can take?"

Hence, the answer is 21.

Choice C is the correct answer.