The table of tangents holds values for every acute angle from 0o to 90o. Each angle has a unique tangent ratio. We get this ratio when two lines meet to make the angle.
Every combination of opposite and adjacent lines that makes a right angled triangle has a unique angle which they make.
If we know the acute angle in a right angled triangle, we can use tables of tangents. This helps us find its corresponding tangent ratio. Similarly, if we know the angle and just one side, we can find the angle’s ratio. Then, we use the tangent relationship to find the other side.
The table of tangents consists of angles from 0o to 90o. We express these angles in 4 significant figures and record their values in a table. All we need to do as mathematician is get a certain angle and find it’s corresponding ratio from the tables.
We expresses angles in the table of tangents in degrees, points of degrees and as well as in minutes. 1 degree (1o) is equivalent to 60 minutes(60′).
We have divided the table of tangents into three major columns as shown in the table extract below:
The first column represents whole number degrees from 0o to 90o and has column head labeled xo which represents
The second column consists of 0.0o to 0.9o which divides a degree into 10 smaller units hence giving an accuracy of 0.1o.
The third column is the one we have labeled ADD and it provides the second decimal value of the angle. Using the table of tangents, we can find angles u to second decimal places.
Example
Determine the tangent of 36.57o
solution
In the column labelled xo , look for the row headed 36 and then move along this row until you reach 0.5. The number at the intersection of 36 and 0.5 is 0.7400
note that the number is recorded as 7400 and not 0.7400. This is done to save on space but you should check the first column after 36, That is, column headed 0.0, whatever value that is stated on that row in that column should be used as the starting value for all the columns in that row.
so tan 36.5 =0.7400, to get the value for tan 36.57, we go to the add column and check on the column 0.07 and add it’s value on the far right of our previous value we read from the table. In this case it is 19 and should be read as 0.0019
hence tan 36.57 should be 0.7400+0.0019 = 0.7419
Example
Use tables to find the tangent of 77o48′
solution
1o=60′, hence 48′ = (48′ x 1o)/60′ = 0.8o
then 77o48′ can be expressed as 77.8o
From the tables, you identify row 77 at xo column then move up to to the column 0.8 and read off that value at the intersection. This value is 0.6252 hence tan 77o48′ = tan 77.8o = 0.6252
Here are exam questions on waves that are common in national exams.
State two differences between electromagnetic waves and mechanical waves (2 marks)
Figure 3 show straight waves incident on a divergent lens placed in a ripple tank to reduce its depth.
Complete the diagram to show the waves in both the shallow region and beyond the lens (2 marks)
3. A ship in an ocean sends out an ultra sound whose echo is received after 3 seconds. if the wavelength of the ultra sound in water is 7.5 cm and the frequency of the transmitter is 20 kHz, determine the depth of the ocean. (3 marks)
4. Explain the fact that radiant heat from the sun penetrates a glass sheet while radian heat from burning wood is cut off by the glass sheet. (2 marks)
Question 5
5. (a) figure 5 shows a displacement-time graph for a progressive wave.
figure 5
(i) State the amplitude of the wave (1 mark)
(ii)Determine the frequency of the wave (4 marks)
(iii) Given that the velocity of the wave is 20 ms-1 , determine it’s wavelength. (3 marks)
(b)Figure 6 shows two identical dippers A and B vibrating in water in phase with each other . The dippers have the same constant frequency and amplitude. The waves produced are observed along the line MN:
Figure 6
It is observed that the amplitude are maximum at points Q and S and minimum at points P and R.
(i) Explain why the amplitude is maximum at Q. (2 marks)
(ii) state why the amplitude is minimum at R (1 mark)
(iii) State what would have happen if the two dippers had different frequencies . ( 1 mark)
6. Figure 7 shows water waves incident on a shallow region of the shape shown with dotted line.
Figure 7
On the same diagram, sketch the wave pattern in and beyond the shallow region (1 mark)
7 . Figure 7 shows standing wave on a string. It is drawn to a scale of 1:5
Figure 7
(a) Indicate on the diagram the wavelength of the standing wave (1 mark)
(b) Determine the wavelength of the wave. (1 mark)
The Grade 9 to Senior School selection process in Kenya involves students choosing their preferred pathways. They also choose subject combinations and schools. This is done through an online system managed by the Ministry of Education.
This process is part of the transition to Senior School under the Competency-Based Education (CBE) framework. Students will select their pathways and subject combinations and they will also choose up to 12 schools across four clusters. STEM is a mandatory pathway.
SELECTION OF PATHWAYS AND SENIOR SCHOOLS
Determination of pathways per senior school
Determination of vacancies for boarding and day schooling in senior schools
Selection of pathways, subjects’ combination and schools by grade 9 learners
Selection based on pathway
The learner will select 12 schools for their chosen pathway as follows.
4 schools in first choice track and subject combination
Four (4) schools in second choice subject combination
Four (4) schools in third choice subject combination (Total 12 schools)
Selection based on accommodation
Out of the 12 schools selected based on pathway:
9 will be boarding schools; 3 from the learners’ home county, 6 from outside their home county/county of residence.
Three (3) day schools in their home sub county/sub county of residence. (Total 12 schools) Pre selection – A school that does not allow open placement can apply to be pre-select if it meets the criteria defined by the Ministry of Education.
Accommodation- Based Breakdown
Top 6 learners per gender in each STEM track per sub-county will be placed for Boarding in schools of choice
Top 3 learners per gender in each Social Science track per sub-county will be placed for Boarding in schools of choice
Top 2 learners per gender in each Arts and Sports Science track per sub-county be placed to Boarding schools of their choice
Placement of Candidates with Achievement Level of averaging 7 and 8 per track to boarding schools of their choice
The Fundamental Theorem of Calculus establishes a crucial link between differentiation and integration.
It essentially states that these two operations are inverses of each other, and it provides a way to evaluate definite integrals using anti-derivatives.
Suppose that f is continuous at a closed interval [a, b] . If the function F is defined on a closed interval [a, b] by:
$$F(x) = \int_{a}^{x} f(t) dt $$
where a is a real number, Then F is the anti-derivative of f. in other words, F'(x) = f(x)
consider the relationships:
then
f(x) = x2and
Note: We use the dummy variable (t) in the integrand to avoid confusion with the upper limit x.
Sometimes the fundamental theorem of calculus is interpreted to mean that:
differentiation and integration are inverse processes to each other.
It follows that:
The fundamental theorem of calculus states that:
if f is continous on an open interval containing a and x and then we first integrate the function f and then differentiate with respect to x, then the result we get is the function f again.
In other words, the fundamental theorem of calculus argues that differentiation cancels the effect of intergration of continous f(x’).
in short:
For example
Example problem1
Use the fundamental theorem of calculus to find derivative of the following functions
(a)
solution
NOTE: The best way to benefit from this examples is trying the problem first before looking for answers and attempting again after checking your work against the answer.
Example problem2
(b)
solution to problem 2
Example problem 3
Find h'(x) given that :
solution
let y=h(x) and u=x2 and hence:
since u=x2;
and therefore:
By use of chain rule:
which implies u3sinu(2x) = (x2)3sin(x2)2x resulting to:
=2x7sin(x2)
Example problem 4
Consider the expression below, we exchange the limits in the intergral and then change the sign from positive to negative before using the fundamental theorem to solve it.
Example problems on fundamental theorem of calculus
We exchange limits and so the sign of the integral so that the upper limit is the valuable x.
Example problem 6
Use the fundamental theorem of calculus to solve:
Solution
splitting the integral about point zero we have:
and then exchanging limits in the first integral;
let u=-x; first part of the expression above becomes;
from laws of differentiation du/dx=-1 and using chain rule;
The most common calculator used by high school learners is ms fx-82 model. Calculator has many scientific functions that can help a student work out many scientific computations. However, ms fx-82 calculator is non-programmable.
Casio fx-82MS Scientific Calculator
The Casio fx-82 MS is a widely used scientific calculator. It is especially favored by students and professionals for its reliability. It also complies with exam requirements.
Key Features of fx-82 calculator:
240 Functions: Includes trigonometric, statistical, fractional, and exponential calculations.
Natural Textbook Display: Shows expressions as they appear in textbooks for easy comprehension.
Two-Line Display: Simultaneously displays input and output for clarity.
Multi-Replay Function: Allows quick recall and editing of previous formulas.
STAT-Data Editor: Supports mean, standard deviation, and regression analysis.
9 Variable Memories: Stores and recalls up to 9 data sets for efficiency.
Durable Design: Features robust plastic keys and a protective slide-on hard case.
Battery Powered: Operates on a single AAA battery.
Non-Programmable: Compliant with exam standards for academic use.
Portable and Lightweight: Compact design for easy everyday use.
Handling Precautions
Even if the calculator is operating normally, replace the battery at least after every two years. Continued use after the specified number of years can result to abnormal operation.
You should replace the battery immediately after display figures become dim.
A dead battery can leak, causing damage to and malfunction of the calculator. Never leave a dead battery in the calculator.
The battery that comes with the calculator is for factory testing, and it discharges slightly during shipment and storage. Because of these reasons, its battery life can be shorter than normal. For that reason, consider replacing the battery sooner.
Avoid use and storage of the calculator in areas subjected to temperature extremes, and large amounts of humidity and dust.
Do not subject the calculator to excessive impact, pressure, or bending.
Never try to take the calculator apart.
Use a soft, dry cloth to clean the exterior of the calculator
Do not use a nickel-based primary battery with this product.
use of incompatible batteries such as nickel-based primary battery with fx-82ms calculator can result in shorter battery life and product malfunctioning.
press 2 and the use > and < to adjust display contrast. when satisfied with the display settings, press AC button
use of calculator keys
To use the alternate function of a key, press [SHIFT] key followed by the key. The alternate function is indicated by the text printed above the key. The alternate function is usually marked with a different color from the main key.
Basic operations in calculator
use AC button to clear all values.
To clear memory press shift then mode . Three screens will display as follow:
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