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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.

## 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 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 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.

## 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

## 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

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

A score between 0 and 1 means: ?

A score of 2 means: ?

A score of 3 means: ?

A score of 4 means: ?