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Revise Trigonometry for Sat Mathematics.

An expert in teaching high school mathematics with three years teaching experience.

Introduction to Trigonometry.

The word 'Trigonometry' is derived from the two Greek words, 'Trigonon' meaning 'Triangles' and 'Metron' meaning 'Measurement'.


So in simple words, Trigonometry means the measurement of triangles.

Consider the right triangle PQR below. The longest side RQ, is called the hypotenuse. One of the two shorter sides, PR is opposite to the indicated acute angle

revise-trigonometry-for-sat-mathematics

The Trigonometric Ratios.

In the above diagram, the sine of the angle alpha is the ratio of the opposite side to the hypotenuse.


The cosine of the angle alpha is the ratio of the adjacent sides to the hypotenuse.


The tangent of the angle alpha is the ratio of the opposite side to the adjacent side .



We use the mnemonic SOH CAH TOA , to recollect these ratios .

The cosecant ratio is the reciprocal of the sine ratio, the secant ratio is the reciprocal of the cosine ratio, and the cotangent ratio is the reciprocal of t

revise-trigonometry-for-sat-mathematics

The right triangle PQR below has sides of length as indicated. Find a) sin(Q) b) cos(Q) c) tan(Q) d) sin (R) e) cos(R) f) tan(R).

revise-trigonometry-for-sat-mathematics

In right triangle PQR below, tan(Q) = ¾. Find the values of the other five trigonometric ratios.

revise-trigonometry-for-sat-mathematics

The six trigonometric ratios are shown in the table below

Trig RatiosReciprocals

Sine

Cosecant

Cosine

Secant

Tangent

Cotangent

The trigonometric ratios for special angles

The trigonometric ratios for most angles are not exact. There are few ones that are exact. These include the trigonometric ratios for 30, 45, 60 and 90 degrees. The diagram below illustrates the exact values for these special angles.

The trigonometric ratio for 30 and 60 degrees angles

revise-trigonometry-for-sat-mathematics
revise-trigonometry-for-sat-mathematics

Trigonometric ratios for 45 degrees angle

revise-trigonometry-for-sat-mathematics

The trigonometric ratios for 0° and 90° angles.

We can also use similar approaches to derive some few more exact values for the 0° and 90° angle.


For instance, from the 45°-45°-90° triangle above, as the height of the triangle diminishes to zero, the hypotenuse equals the adjacent side.

Therefore the sine of the zero degree angle is 0, since the opposite side has reduced to zero.

That is :

sin(0°)=0

The cosine of the zero degree angle will be 1. Since the adjacent side is now equal to the hypotenuse.

That is :

cos(0°)=1

It follows that the tangent of the 0° angle is 0. That is:

tan(0°)=0, since tan(0°)=sin(0°)/cos(0°).


Or using the same argument that the opposite side is now zero.

Using the same argument, we can further establish that:

sin(90°)=1

cos(90°)=0

and

tan(90°) is not defined.


Consider the right triangle below. We can establish some useful relations among the trigonometric ratios using the Pythagoras theorem.

revise-trigonometry-for-sat-mathematics

The Pythagorean Identity.

In the above right triangle, the Pythagoras theorem tells us that:

a² + b² = c² .. .(1)


Dividing through by c²

We obtain;

(a/c)² + (b/c)² = 1

But;

sin(x) =a/c

cos(x)=b/c

On substitution, We obtain;

sin²x + cos²(x) = 1. ... (2)

Similarly, we can show that;

tan²x +1 = sec²x

cot²x + 1 = csc²x


These three identities are referred to as the Pythagorean identities. The reason is that we derived them using the Pythagoras Theorem.


Worked Example 4


In a given triangle, sin(x) = 0.6. What is the value of:

i. cos(x)

ii.tan(x)

If x is an acute angle.


Solution


sin²x + cos²(x) = 1

(0.6)² + cos²(x) = 1

cos²(x) = 1- (0.6)²


cos²(x) = 1- 0.36

cos²(x) = 0.64

cos(x)=±√(0.64)

cos(x) = 0.8, since x is acute.

ii. tan(x) = 0.6/0.8

tan(x) = 0.75






Find the exact values of x and y in the diagram below

revise-trigonometry-for-sat-mathematics

Solution to worked example 5 on exact values.

The 45° has the opposite side given and the variable x is on the hypotenuse. We can find the value of x using the sine ratio.


sin(45°)=4/x


√2/2=4/x


When we cross multiply we get,


x√2 = 4×2

x√2 = 8


Dividing both sides by √2 gives

x=8/√2

Rationalizing the denominator gives


x= (8√2)/2

This implies that

x= 4√2


The 30° angle has the opposite side given and the variable y is on the adjacent side. We can find the value of y using the tangent ratio.


tan(30°)= 4/y


1/√3 =4/y


When we reciprocate both sides we obtain

√3 = y/4


Making y the subject gives


y=4√3



Worked Example 6.

In triangle ABC, <B=90°, the length of the hypotenuse, AC is 10cm, tan(C) =¾. Find the value of sec(C) if 0°≤C<90°


Solution

Using the Pythagorean identity,

tan²(C°)+1= sec²(C°)


We can easily find the value of sec(C).


So let's substitute


tan(C)= ¾

into the above relation.


(¾)² +1=se²(C)


sec²(C) = (9/16) +1


sec²(C) = 25/16


Taking the square root of both sides gives

sec(C)=±√(25/16)

sec(C)= -5/4 or 5/4

Since <C is acute,

sec(C) = 5/4


More examples will be added soon

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SAT Math Level 2


SAT Level 1 Trigonometry Practice Questions

For each question, choose the best answer for you.

  1. What is the base angle of an isosceles right triangle in degrees?
    • 30
    • 40
    • 25
    • 45
    • cannot be determined
  2. The hypotenuse of an isosceles right triangle measures x units. What is the length of a base of the triangle?
    • x√2/2 units
    • x units
    • √x/2 units
    • ½x units
    • x√2 units
    • x² units
  3. If cos(x°) = 0.75. Find csc(x°).
    • ¾
    • ⁴/3
    • ¼
  4. If the longer base of triangle XYZ is 3 times the shorter base,a, what is the length of the hypotenuse in terms of a?
    • a√2
    • a√3
    • a√10
    • 3a
  5. Given that sec²(x)= a, find tan²(x).
    • a² -1
    • 1+a²
    • 1-a
    • a-1

Scoring

Use the scoring guide below to add up your total points based on your answers.

  1. What is the base angle of an isosceles right triangle in degrees?
    • 30: +0 points
    • 40: +0 points
    • 25: +0 points
    • 45: +1 point
    • cannot be determined: +0 points
  2. The hypotenuse of an isosceles right triangle measures x units. What is the length of a base of the triangle?
    • x√2/2 units: +0 points
    • x units: +0 points
    • √x/2 units: +0 points
    • ½x units: +0 points
    • x√2 units: +1 point
    • x² units: +0 points
  3. If cos(x°) = 0.75. Find csc(x°).
    • ¾: +0 points
    • ⅝: +0 points
    • ⅜: +0 points
    • ⁴/3: +1 point
    • ¼: +0 points
  4. If the longer base of triangle XYZ is 3 times the shorter base,a, what is the length of the hypotenuse in terms of a?
    • a√2: +0 points
    • a√3: +0 points
    • a√10: +0 points
    • 3a: +0 points
    • a²: +0 points
  5. Given that sec²(x)= a, find tan²(x).
    • a²: +0 points
    • a² -1: +0 points
    • 1+a²: +0 points
    • 1-a: +0 points
    • a-1: +1 point

Interpreting Your Score

A score between 0 and 1 means: ?

A score of 2 means: ?

A score of 3 means: ?

A score of 4 means: ?

Comments

George Dimitriadis from Templestowe on October 01, 2018:

Hi.

A useful article, but perhaps some examples at useful intervals will reinforce the concepts. Also, some minor grammatical errors can be corrected.

In the quiz, I think the answer to question 3 should be 4/sqrt(7), and the answer to question 5 is a-1, not x-1.

Regards

George