ਸਿੱਖਣ ਦੇ ਉਦੇਸ਼
ਇਸ ਭਾਗ ਵਿੱਚ, ਤੁਸੀਂ:
π 3 , π 3 , π 4 , π 4 , ਅਤੇ π 6 . π 6 . ਦੇ ਤ੍ਰਿਕੋਣਮਿਤੀ ਫਲਨ ਸੈਕੰਟ, ਕੋਸੈਕੰਟ, ਟੈਂਜੈਂਟ, ਅਤੇ ਕੋਟੈਂਜੈਂਟ ਦੇ ਸਹੀ ਮੁੱਲ ਪਾਓਗੇ।
ਹਵਾਲਾ ਕੋਣਾਂ ਦੀ ਵਰਤੋਂ ਕਰਕੇ ਤ੍ਰਿਕੋਣਮਿਤੀ ਫਲਨ ਸੈਕੰਟ, ਕੋਸੈਕੰਟ, ਟੈਂਜੈਂਟ, ਅਤੇ ਕੋਟੈਂਜੈਂਟ ਦਾ ਮੁੱਲ ਪਤਾ ਕਰੋ।
ਸਮ ਅਤੇ ਟਾਂਕ ਤ੍ਰਿਕੋਣਮਿਤੀ ਫਲਨਾਂ ਦੇ ਗੁਣਾਂ ਦੀ ਵਰਤੋਂ ਕਰੋ।
ਮੁਢਲੀਆਂ ਪਛਾਣਾਂ ਨੂੰ ਪਛਾਣੋ ਅਤੇ ਵਰਤੋਂ।
ਕੈਲਕੂਲੇਟਰ ਨਾਲ ਤ੍ਰਿਕੋਣਮਿਤੀ ਫਲਨਾਂ ਦਾ ਮੁੱਲ ਪਤਾ ਕਰੋ।
ਅਮਰੀਕੀ ਵਿਕਲਾਂਗ ਐਕਟ ਦੇ ਮਾਪਦੰਡਾਂ ਨੂੰ ਪੂਰਾ ਕਰਨ ਵਾਲਾ ਇੱਕ ਵ੍ਹੀਲਚੇਅਰ ਰੈਂਪ, ਜ਼ਮੀਨ ਨਾਲ ਅਜਿਹਾ ਕੋਣ ਬਣਾਉਣਾ ਚਾਹੀਦਾ ਹੈ ਜਿਸਦਾ ਟੈਂਜੈਂਟ 1 12 1 12 ਜਾਂ ਇਸ ਤੋਂ ਘੱਟ ਹੋਵੇ, ਭਾਵੇਂ ਉਸਦੀ ਲੰਬਾਈ ਕੁਝ ਵੀ ਹੋਵੇ। ਟੈਂਜੈਂਟ ਇੱਕ ਅਨੁਪਾਤ ਦਰਸਾਉਂਦਾ ਹੈ, ਇਸ ਲਈ ਇਸਦਾ ਮਤਲਬ ਹੈ ਕਿ ਹਰ 1 ਇੰਚ ਚੜ੍ਹਾਈ ਲਈ, ਰੈਂਪ ਵਿੱਚ 12 ਇੰਚ ਦੌੜ ਹੋਣੀ ਚਾਹੀਦੀ ਹੈ। ਤ੍ਰਿਕੋਣਮਿਤੀ ਫਲਨ ਸਾਨੂੰ ਸਹੀ ਮਾਪਾਂ ਤੋਂ ਸੁਤੰਤਰ ਵਸਤੂਆਂ ਦੇ ਆਕਾਰ ਅਤੇ ਅਨੁਪਾਤ ਨਿਰਧਾਰਤ ਕਰਨ ਦੀ ਆਗਿਆ ਦਿੰਦੇ ਹਨ। ਅਸੀਂ ਪਹਿਲਾਂ ਹੀ ਇੱਕ ਕੋਣ ਦੇ ਸਾਈਨ ਅਤੇ ਕੋਸਾਈਨ ਫਲਨਾਂ ਨੂੰ ਪਰਿਭਾਸ਼ਿਤ ਕਰ ਚੁੱਕੇ ਹਾਂ। ਭਾਵੇਂ ਸਾਈਨ ਅਤੇ ਕੋਸਾਈਨ ਸਭ ਤੋਂ ਵੱਧ ਵਰਤੇ ਜਾਣ ਵਾਲੇ ਤ੍ਰਿਕੋਣਮਿਤੀ ਫਲਨ ਹਨ, ਚਾਰ ਹੋਰ ਵੀ ਹਨ। ਇਕੱਠੇ ਮਿਲ ਕੇ, ਉਹ ਛੇ ਤ੍ਰਿਕੋਣਮਿਤੀ ਫਲਨਾਂ ਦਾ ਸਮੂਹ ਬਣਾਉਂਦੇ ਹਨ। ਇਸ ਭਾਗ ਵਿੱਚ, ਅਸੀਂ ਬਾਕੀ ਫਲਨਾਂ ਦੀ ਜਾਂਚ ਕਰਾਂਗੇ।
ਤ੍ਰਿਕੋਣਮਿਤੀ ਫਲਨਾਂ ਸੈਕੰਟ, ਕੋਸੈਕੰਟ, ਟੈਂਜੈਂਟ, ਅਤੇ ਕੋਟੈਂਜੈਂਟ ਦੇ ਸਹੀ ਮੁੱਲ ਪਾਉਣਾ
ਬਾਕੀ ਫਲਨਾਂ ਨੂੰ ਪਰਿਭਾਸ਼ਿਤ ਕਰਨ ਲਈ, ਅਸੀਂ ਇੱਕ ਵਾਰ ਫਿਰ ਇੱਕ ਇਕਾਈ ਚੱਕਰ ਬਣਾਵਾਂਗੇ ਜਿਸ ਵਿੱਚ t, t ਦੇ ਕੋਣ ਦੇ ਅਨੁਸਾਰੀ ਇੱਕ ਬਿੰਦੂ ( x,y ) ( x,y ) ਹੋਵੇ, ਜਿਵੇਂ ਕਿ ਚਿੱਤਰ 1 ਵਿੱਚ ਦਿਖਾਇਆ ਗਿਆ ਹੈ। ਸਾਈਨ ਅਤੇ ਕੋਸਾਈਨ ਵਾਂਗ, ਅਸੀਂ ਦੂਜੇ ਫਲਨਾਂ ਨੂੰ ਲੱਭਣ ਲਈ ( x,y ) ( x,y ) ਨਿਰਦੇਸ਼ਾਂਕਾਂ ਦੀ ਵਰਤੋਂ ਕਰ ਸਕਦੇ ਹਾਂ।
ਪਹਿਲਾ ਫਲਨ ਜੋ ਅਸੀਂ ਪਰਿਭਾਸ਼ਿਤ ਕਰਾਂਗੇ ਉਹ ਟੈਂਜੈਂਟ ਹੈ। ਇੱਕ ਕੋਣ ਦਾ ਟੈਂਜੈਂਟ ਇਕਾਈ ਚੱਕਰ ਉੱਤੇ ਅਨੁਸਾਰੀ ਬਿੰਦੂ ਦੇ y-ਮੁੱਲ ਅਤੇ x-ਮੁੱਲ ਦਾ ਅਨੁਪਾਤ ਹੈ। ਚਿੱਤਰ 1 ਵਿੱਚ, ਕੋਣ t t ਦਾ ਟੈਂਜੈਂਟ y x ,x≠0. y x ,x≠0 ਦੇ ਬਰਾਬਰ ਹੈ। ਕਿਉਂਕਿ y-ਮੁੱਲ t t ਦੇ ਸਾਈਨ ਦੇ ਬਰਾਬਰ ਹੈ, ਅਤੇ x-ਮੁੱਲ t t ਦੇ ਕੋਸਾਈਨ ਦੇ ਬਰਾਬਰ ਹੈ, ਕੋਣ t t ਦਾ ਟੈਂਜੈਂਟ ਵੀ sint cost ,cost≠0. sint cost ,cost≠0 ਦੇ ਤੌਰ ਤੇ ਪਰਿਭਾਸ਼ਿਤ ਕੀਤਾ ਜਾ ਸਕਦਾ ਹੈ। ਟੈਂਜੈਂਟ ਫਲਨ ਨੂੰ tan. tan. ਦੇ ਰੂਪ ਵਿੱਚ ਸੰਖੇਪ ਕੀਤਾ ਜਾਂਦਾ ਹੈ। ਬਾਕੀ ਤਿੰਨ ਫਲਨਾਂ ਨੂੰ ਪਹਿਲਾਂ ਹੀ ਪਰਿਭਾਸ਼ਿਤ ਕੀਤੇ ਗਏ ਫਲਨਾਂ ਦੇ ਉਲਟੇ ਵਜੋਂ ਪ੍ਰਗਟ ਕੀਤਾ ਜਾ ਸਕਦਾ ਹੈ।
ਸੈਕੰਟ ਫਲਨ ਕੋਸਾਈਨ ਫਲਨ ਦਾ ਉਲਟਾ ਹੈ। ਚਿੱਤਰ 1 ਵਿੱਚ, ਕੋਣ t t ਦਾ ਸੈਕੰਟ 1 cost = 1 x ,x≠0. 1 cost = 1 x ,x≠0 ਦੇ ਬਰਾਬਰ ਹੈ। ਸੈਕੰਟ ਫਲਨ ਨੂੰ sec. sec. ਦੇ ਰੂਪ ਵਿੱਚ ਸੰਖੇਪ ਕੀਤਾ ਜਾਂਦਾ ਹੈ।
ਕੋਟੈਂਜੈਂਟ ਫਲਨ ਟੈਂਜੈਂਟ ਫਲਨ ਦਾ ਉਲਟਾ ਹੈ। ਚਿੱਤਰ 1 ਵਿੱਚ, ਕੋਣ t t ਦਾ ਕੋਟੈਂਜੈਂਟ cost sint = x y ,y≠0. cost sint = x y ,y≠0 ਦੇ ਬਰਾਬਰ ਹੈ। ਕੋਟੈਂਜੈਂਟ ਫਲਨ ਨੂੰ cot. cot. ਦੇ ਰੂਪ ਵਿੱਚ ਸੰਖੇਪ ਕੀਤਾ ਜਾਂਦਾ ਹੈ।
ਕੋਸੈਕੰਟ ਫਲਨ ਸਾਈਨ ਫਲਨ ਦਾ ਉਲਟਾ ਹੈ। ਚਿੱਤਰ 1 ਵਿੱਚ, ਕੋਣ t t ਦਾ ਕੋਸੈਕੰਟ 1 sint = 1 y ,y≠0. 1 sint = 1 y ,y≠0 ਦੇ ਬਰਾਬਰ ਹੈ। ਕੋਸੈਕੰਟ ਫਲਨ ਨੂੰ csc. csc. ਦੇ ਰੂਪ ਵਿੱਚ ਸੰਖੇਪ ਕੀਤਾ ਜਾਂਦਾ ਹੈ।
ਟੈਂਜੈਂਟ, ਸੈਕੰਟ, ਕੋਸੈਕੰਟ, ਅਤੇ ਕੋਟੈਂਜੈਂਟ ਫਲਨ
ਜੇਕਰ t t ਇੱਕ ਅਸਲ ਸੰਖਿਆ ਹੈ ਅਤੇ (x,y) (x,y) ਇੱਕ ਬਿੰਦੂ ਹੈ ਜਿੱਥੇ t t ਰੇਡੀਅਨ ਦੇ ਕੋਣ ਦੀ ਅੰਤਿਮ ਭੁਜਾ ਇਕਾਈ ਚੱਕਰ ਨੂੰ ਕੱਟਦੀ ਹੈ, ਤਾਂ
ਉਦਾਹਰਨ 1
ਇਕਾਈ ਚੱਕਰ ਉੱਤੇ ਇੱਕ ਬਿੰਦੂ ਤੋਂ ਤ੍ਰਿਕੋਣਮਿਤੀ ਫਲਨਾਂ ਦਾ ਪਤਾ ਲਗਾਉਣਾ
ਬਿੰਦੂ ( − 3 2 , 1 2 ) ( − 3 2 , 1 2 ) ਇਕਾਈ ਚੱਕਰ ਉੱਤੇ ਹੈ, ਜਿਵੇਂ ਕਿ ਚਿੱਤਰ 2 ਵਿੱਚ ਦਿਖਾਇਆ ਗਿਆ ਹੈ। sint,cost,tant,sect,csct, sint,cost,tant,sect,csct, ਅਤੇ cott. cott. ਦਾ ਪਤਾ ਲਗਾਓ।
ਹੱਲ
ਕਿਉਂਕਿ ਅਸੀਂ ਕੋਣ t t ਦੁਆਰਾ ਦਰਸਾਏ ਗਏ ਇਕਾਈ ਚੱਕਰ ਉੱਤੇ ਬਿੰਦੂ ਦੇ (x,y) (x,y) ਨਿਰਦੇਸ਼ਾਂਕ ਜਾਣਦੇ ਹਾਂ, ਅਸੀਂ ਛੇ ਫਲਨਾਂ ਨੂੰ ਲੱਭਣ ਲਈ ਉਨ੍ਹਾਂ ਨਿਰਦੇਸ਼ਾਂਕਾਂ ਦੀ ਵਰਤੋਂ ਕਰ ਸਕਦੇ ਹਾਂ:
ਇਸਨੂੰ ਅਜ਼ਮਾਓ #1
ਬਿੰਦੂ ( 2 2 ,− 2 2 ) ( 2 2 ,− 2 2 ) ਇਕਾਈ ਚੱਕਰ ਉੱਤੇ ਹੈ, ਜਿਵੇਂ ਕਿ ਚਿੱਤਰ 3 ਵਿੱਚ ਦਿਖਾਇਆ ਗਿਆ ਹੈ। sint,cost,tant,sect,csct, sint,cost,tant,sect,csct, ਅਤੇ cott. cott. ਦਾ ਪਤਾ ਲਗਾਓ।
ਉਦਾਹਰਨ 2
Finding the Trigonometric Functions of an Angle
Find sint,cost,tant,sect,csct, sint,cost,tant,sect,csct, and cott cott when t= π 6 . t= π 6 .
Solution
We have previously used the properties of equilateral triangles to demonstrate that sin π 6 = 1 2 sin π 6 = 1 2 and cos π 6 = 3 2 . cos π 6 = 3 2 . We can use these values and the definitions of tangent, secant, cosecant, and cotangent as functions of sine and cosine to find the remaining function values.
Try It #2
Find sint,cost,tant,sect,csct, sint,cost,tant,sect,csct, and cott cott when t= π 3 . t= π 3 .
Because we know the sine and cosine values for the common first-quadrant angles, we can find the other function values for those angles as well by setting x x equal to the cosine and y y equal to the sine and then using the definitions of tangent, secant, cosecant, and cotangent. The results are shown in Table 1.
row: Angle | 0 0 | π 6 , or 30° π 6 , or 30° | π 4 , or 45° π 4 , or 45° | π 3 , or 60° π 3 , or 60° | π 2 , or 90° π 2 , or 90°
row: Cosine | 1 | 3 2 3 2 | 2 2 2 2 | 1 2 1 2 | 0
row: Sine | 0 | 1 2 1 2 | 2 2 2 2 | 3 2 3 2 | 1
row: Tangent | 0 | 3 3 3 3 | 1 | 3 3 | Undefined
row: Secant | 1 | 2 3 3 2 3 3 | 2 2 | 2 | Undefined
row: Cosecant | Undefined | 2 | 2 2 | 2 3 3 2 3 3 | 1
row: Cotangent | Undefined | 3 3 | 1 | 3 3 3 3 | 0
Using Reference Angles to Evaluate Tangent, Secant, Cosecant, and Cotangent
We can evaluate trigonometric functions of angles outside the first quadrant using reference angles as we have already done with the sine and cosine functions. The procedure is the same: Find the reference angle formed by the terminal side of the given angle with the horizontal axis. The trigonometric function values for the original angle will be the same as those for the reference angle, except for the positive or negative sign, which is determined by x- and y-values in the original quadrant. Figure 4 shows which functions are positive in which quadrant.
To help us remember which of the six trigonometric functions are positive in each quadrant, we can use the mnemonic phrase “A Smart Trig Class.” Each of the four words in the phrase corresponds to one of the four quadrants, starting with quadrant I and rotating counterclockwise. In quadrant I, which is “A,” underlineaend underlinell of the six trigonometric functions are positive. In quadrant II, “Smart,” only underlinesend underlineine and its reciprocal function, cosecant, are positive. In quadrant III, “Trig,” only underlinetend underlineangent and its reciprocal function, cotangent, are positive. Finally, in quadrant IV, “Class,” only underlinecend underlineosine and its reciprocal function, secant, are positive.
How To
Given an angle not in the first quadrant, use reference angles to find all six trigonometric functions.
Measure the angle formed by the terminal side of the given angle and the horizontal axis. This is the reference angle.
Evaluate the function at the reference angle.
Observe the quadrant where the terminal side of the original angle is located. Based on the quadrant, determine whether the output is positive or negative.
Example 3
Using Reference Angles to Find Trigonometric Functions
Use reference angles to find all six trigonometric functions of − 5π 6 . − 5π 6 .
Solution
The angle between this angle’s terminal side and the x-axis is π 6 , π 6 , so that is the reference angle. Since − 5π 6 − 5π 6 is in the third quadrant, where both x x and y y are negative, cosine, sine, secant, and cosecant will be negative, while tangent and cotangent will be positive.
Try It #3
Use reference angles to find all six trigonometric functions of − 7π 4 . − 7π 4 .
Using Even and Odd Trigonometric Functions
To be able to use our six trigonometric functions freely with both positive and negative angle inputs, we should examine how each function treats a negative input. As it turns out, there is an important difference among the functions in this regard.
Consider the function f(x)= x 2 , f(x)= x 2 , shown in Figure 5. The graph of the function is symmetrical about the y-axis. All along the curve, any two points with opposite x-values have the same function value. This matches the result of calculation: (4) 2 = (−4) 2 , (4) 2 = (−4) 2 , (−5) 2 = (5) 2 , (−5) 2 = (5) 2 , and so on. So f(x)= x 2 f(x)= x 2 is an even function, a function such that two inputs that are opposites have the same output. That means f( −x )=f( x ). f( −x )=f( x ).
Figure 5: The function f(x)= x 2 f(x)= x 2 is an even function.
Now consider the function f(x)= x 3 , f(x)= x 3 , shown in Figure 6. The graph is not symmetrical about the y-axis. All along the graph, any two points with opposite x-values also have opposite y-values. So f(x)= x 3 f(x)= x 3 is an odd function, one such that two inputs that are opposites have outputs that are also opposites. That means f( −x )=−f( x ). f( −x )=−f( x ).
Figure 6: The function f(x)= x 3 f(x)= x 3 is an odd function.
We can test whether a trigonometric function is even or odd by drawing a unit circle with a positive and a negative angle, as in Figure 7. The sine of the positive angle is y. y. The sine of the negative angle is −y. The sine function, then, is an odd function. We can test each of the six trigonometric functions in this fashion. The results are shown in Table 2.
row: sint=y sin(−t)=−y sint≠sin(−t) sint=y sin(−t)=−y sint≠sin(−t) | cost=x cos(−t)=x cost=cos(−t) cost=x cos(−t)=x cost=cos(−t) | tan(t)= y x tan(−t)=− y x tant≠tan(−t) tan(t)= y x tan(−t)=− y x tant≠tan(−t)
row: sect= 1 x sec(−t)= 1 x sect=sec(−t) sect= 1 x sec(−t)= 1 x sect=sec(−t) | csct= 1 y csc(−t)= 1 −y csct≠csc(−t) csct= 1 y csc(−t)= 1 −y csct≠csc(−t) | cott= x y cot(−t)= x −y cott≠cot(−t) cott= x y cot(−t)= x −y cott≠cot(−t)
Even and Odd Trigonometric Functions
An even function is one in which f(−x)=f(x). f(−x)=f(x).
An odd function is one in which f(−x)=−f(x). f(−x)=−f(x).
Cosine and secant are even:
Sine, tangent, cosecant, and cotangent are odd:
Example 4
Using Even and Odd Properties of Trigonometric Functions
If the secant of angle t t is 2, what is the secant of −t? −t?
Solution
Secant is an even function. The secant of an angle is the same as the secant of its opposite. So if the secant of angle t is 2, the secant of −t −t is also 2.
Try It #4
If the cotangent of angle t t is 3 , 3 , what is the cotangent of −t? −t?
Recognizing and Using Fundamental Identities
We have explored a number of properties of trigonometric functions. Now, we can take the relationships a step further, and derive some fundamental identities. Identities are statements that are true for all values of the input on which they are defined. Usually, identities can be derived from definitions and relationships we already know. For example, the Pythagorean Identity we learned earlier was derived from the Pythagorean Theorem and the definitions of sine and cosine.
Fundamental Identities
We can derive some useful identities from the six trigonometric functions. The other four trigonometric functions can be related back to the sine and cosine functions using these basic relationships:
Example 5
Using Identities to Evaluate Trigonometric Functions
ⓐ Given sin(45°)= 2 2 ,cos(45°)= 2 2 , sin(45°)= 2 2 ,cos(45°)= 2 2 , evaluate tan(45°). tan(45°).
ⓑ Given sin( 5π 6 )= 1 2 ,cos( 5π 6 )=− 3 2 ,evaluatesec( 5π 6 ). sin( 5π 6 )= 1 2 ,cos( 5π 6 )=− 3 2 ,evaluatesec( 5π 6 ).
Solution
Because we know the sine and cosine values for these angles, we can use identities to evaluate the other functions.
ⓐ tan(45°)= sin(45°) cos(45°) = 2 2 2 2 =1 tan(45°)= sin(45°) cos(45°) = 2 2 2 2 =1
ⓑ sec( 5π 6 )= 1 cos( 5π 6 ) = 1 − 3 2 = −2 3 =− 2 3 3 sec( 5π 6 )= 1 cos( 5π 6 ) = 1 − 3 2 = −2 3 =− 2 3 3
Try It #5
Evaluate csc( 7π 6 ). csc( 7π 6 ).
Example 6
Using Identities to Simplify Trigonometric Expressions
Simplify sect tant . sect tant .
Solution
We can simplify this by rewriting both functions in terms of sine and cosine.
By showing that sect tant sect tant can be simplified to csct, csct, we have, in fact, established a new identity.
Try It #6
Simplify (tant)(cost). (tant)(cost).
Alternate Forms of the Pythagorean Identity
We can use these fundamental identities to derive alternative forms of the Pythagorean Identity, cos 2 t+ sin 2 t=1. cos 2 t+ sin 2 t=1. One form is obtained by dividing both sides by cos 2 t: cos 2 t:
The other form is obtained by dividing both sides by sin 2 t: sin 2 t:
Alternate Forms of the Pythagorean Identity
Example 7
Using Identities to Relate Trigonometric Functions
If cos(t)= 12 13 cos(t)= 12 13 and t t is in quadrant IV, as shown in Figure 8, find the values of the other five trigonometric functions.
Solution
We can find the sine using the Pythagorean Identity, cos 2 t+ sin 2 t=1, cos 2 t+ sin 2 t=1, and the remaining functions by relating them to sine and cosine.
The sign of the sine depends on the y-values in the quadrant where the angle is located. Since the angle is in quadrant IV, where the y-values are negative, its sine is negative, − 5 13 . − 5 13 .
The remaining functions can be calculated using identities relating them to sine and cosine.
Try It #7
If sec(t)=− 17 8 sec(t)=− 17 8 and 0<t<π, 0<t<π, find the values of the other five functions.
As we discussed in the chapter opening, a function that repeats its values in regular intervals is known as a periodic function. The trigonometric functions are periodic. For the four trigonometric functions, sine, cosine, cosecant and secant, a revolution of one circle, or 2π, 2π, will result in the same outputs for these functions. And for tangent and cotangent, only a half a revolution will result in the same outputs.
Other functions can also be periodic. For example, the lengths of months repeat every four years. If x x represents the length time, measured in years, and f(x) f(x) represents the number of days in February, then f(x+4)=f(x). f(x+4)=f(x). This pattern repeats over and over through time. In other words, every four years (except for multiples of 100), February typically has the same number of days as it did 4 years earlier. The positive number 4 is the smallest positive number that satisfies this condition and is called the period. A period is the shortest interval over which a function completes one full cycle—in this example, the period is 4 and represents the time it takes for us to be certain February has the same number of days.
Period of a Function
The period P P of a repeating function f f is the number representing the interval such that f(x+P)=f(x) f(x+P)=f(x) for any value of x. x.
The period of the cosine, sine, secant, and cosecant functions is 2π. 2π.
The period of the tangent and cotangent functions is π. π.
Example 8
Finding the Values of Trigonometric Functions
Find the values of the six trigonometric functions of angle t t based on Figure 9.
Solution
Try It #8
Find the values of the six trigonometric functions of angle t t based on Figure 10.
Example 9
Finding the Value of Trigonometric Functions
If sin( t )=− 3 2 sin( t )=− 3 2 and cos(t)= 1 2 , cos(t)= 1 2 , find sec(t),csc(t),tan(t), cot(t). sec(t),csc(t),tan(t), cot(t).
Solution
Try It #9
If sin( t )= 2 2 sin( t )= 2 2 and cos( t )= 2 2 , cos( t )= 2 2 , find sec(t),csc(t),tan(t), and cot(t). sec(t),csc(t),tan(t), and cot(t).
Evaluating Trigonometric Functions with a Calculator
We have learned how to evaluate the six trigonometric functions for the common first-quadrant angles and to use them as reference angles for angles in other quadrants. To evaluate trigonometric functions of other angles, we use a scientific or graphing calculator or computer software. If the calculator has a degree mode and a radian mode, confirm the correct mode is chosen before making a calculation.
Evaluating a tangent function with a scientific calculator as opposed to a graphing calculator or computer algebra system is like evaluating a sine or cosine: Enter the value and press the TAN key. For the reciprocal functions, there may not be any dedicated keys that say CSC, SEC, or COT. In that case, the function must be evaluated as the reciprocal of a sine, cosine, or tangent.
If we need to work with degrees and our calculator or software does not have a degree mode, we can enter the degrees multiplied by the conversion factor π 180 π 180 to convert the degrees to radians. To find the secant of 30°, 30°, we could press
or
How To
Given an angle measure in radians, use a scientific calculator to find the cosecant.
If the calculator has degree mode and radian mode, set it to radian mode.
Enter: 1 / 1 /
Enter the value of the angle inside parentheses.
Press the SIN key.
Press the = key.
How To
Given an angle measure in radians, use a graphing utility/calculator to find the cosecant.
If the graphing utility has degree mode and radian mode, set it to radian mode.
Enter: 1 / 1 /
Press the SIN key.
Enter the value of the angle inside parentheses.
Press the ENTER key.
Example 10
Evaluating the Cosecant Using Technology
Evaluate the cosecant of 5π 7 . 5π 7 .
Solution
For a scientific calculator, enter information as follows:
Try It #10
Evaluate the cotangent of − π 8 . − π 8 .
Media
Access these online resources for additional instruction and practice with other trigonometric functions.
Determining Trig Function Values
More Examples of Determining Trig Functions
Pythagorean Identities
Trig Functions on a Calculator
Verbal
On an interval of [ 0,2π ), [ 0,2π ), can the sine and cosine values of a radian measure ever be equal? If so, where?
What would you estimate the cosine of π π degrees to be? Explain your reasoning.
For any angle in quadrant II, if you knew the sine of the angle, how could you determine the cosine of the angle?
Describe the secant function.
Tangent and cotangent have a period of π. π. What does this tell us about the output of these functions?
Algebraic
For the following exercises, find the exact value of each expression.
tan π 6 tan π 6
sec π 6 sec π 6
csc π 6 csc π 6
cot π 6 cot π 6
tan π 4 tan π 4
sec π 4 sec π 4
csc π 4 csc π 4
cot π 4 cot π 4
tan π 3 tan π 3
sec π 3 sec π 3
csc π 3 csc π 3
cot π 3 cot π 3
For the following exercises, use reference angles to evaluate the expression.
tan 5π 6 tan 5π 6
sec 7π 6 sec 7π 6
csc 11π 6 csc 11π 6
cot 13π 6 cot 13π 6
tan 7π 4 tan 7π 4
sec 3π 4 sec 3π 4
csc 5π 4 csc 5π 4
cot 11π 4 cot 11π 4
tan 8π 3 tan 8π 3
sec 4π 3 sec 4π 3
csc 2π 3 csc 2π 3
cot 5π 3 cot 5π 3
tan225° tan225°
sec300° sec300°
csc150° csc150°
cot240° cot240°
tan330° tan330°
sec120° sec120°
csc210° csc210°
cot315° cot315°
If sint= 3 4 , sint= 3 4 , and t t is in quadrant II, find cost cost, sectsect, csctcsct, tanttant,cott. cott.
If cost=− 1 3 , cost=− 1 3 , and t t is in quadrant III, find sint,sect,csct,tant,cott. sint,sect,csct,tant,cott.
If tant= 12 5 , tant= 12 5 , and 0≤t< π 2 , 0≤t< π 2 , find sint,cost,sect,csct, sint,cost,sect,csct, and cott. cott.
If sint= 3 2 sint= 3 2 and cost= 1 2 , cost= 1 2 , find sect,csct,tant, sect,csct,tant, and cott. cott.
If sin40°≈0.643 sin40°≈0.643 and cos40°≈0.766 cos40°≈0.766 find sec40°,csc40°,tan40°, sec40°,csc40°,tan40°, and cotand40°. cotand40°.
If sint= 2 2 , sint= 2 2 , what is the sin(−t)? sin(−t)?
If cost= 1 2 , cost= 1 2 , what is the cos(−t)? cos(−t)?
If sect=3.1, sect=3.1, what is the sec(−t)? sec(−t)?
If csct=0.34, csct=0.34, what is the csc(−t)? csc(−t)?
If tant=−1.4, tant=−1.4, what is the tan(−t)? tan(−t)?
If cott=9.23, cott=9.23, what is the cot(−t)? cot(−t)?
Graphical
For the following exercises, use the angle in the unit circle to find the value of the each of the six trigonometric functions.
Technology
For the following exercises, use a graphing calculator to evaluate.
csc 5π 9 csc 5π 9
cot 4π 7 cot 4π 7
sec π 10 sec π 10
tan 5π 8 tan 5π 8
sec 3π 4 sec 3π 4
csc π 4 csc π 4
tan98° tan98°
cot33° cot33°
cot140° cot140°
sec310° sec310°
Extensions
For the following exercises, use identities to evaluate the expression.
If tan( t )≈2.7, tan( t )≈2.7, and sin( t )≈0.94, sin( t )≈0.94, find cos( t ). cos( t ).
If tan( t )≈1.3, tan( t )≈1.3, and cos( t )≈0.61, cos( t )≈0.61, find sin( t ). sin( t ).
If csc( t )≈3.2, csc( t )≈3.2, and cos( t )≈0.95, cos( t )≈0.95, find tan( t ). tan( t ).
If cot( t )≈0.58, cot( t )≈0.58, and cos( t )≈0.5, cos( t )≈0.5, find csc( t ). csc( t ).
Determine whether the function f(x)=2sinxcosx f(x)=2sinxcosx is even, odd, or neither.
Determine whether the function f(x)=3 sin 2 xcosx+secx f(x)=3 sin 2 xcosx+secx is even, odd, or neither.
Determine whether the function f(x)=sinx−2 cos 2 x f(x)=sinx−2 cos 2 x is even, odd, or neither.
Determine whether the function f(x)= csc 2 x+secx f(x)= csc 2 x+secx is even, odd, or neither.
For the following exercises, use identities to simplify the expression.
cscttant cscttant
sect csct sect csct
Real-World Applications
The amount of sunlight in a certain city can be modeled by the function h=15cos( 1 600 d ), h=15cos( 1 600 d ), where h h represents the hours of sunlight, and d d is the day of the year. Use the equation to find how many hours of sunlight there are on February 11, the 42nd day of the year. State the period of the function.
The amount of sunlight in a certain city can be modeled by the function h=16cos( 1 500 d ), h=16cos( 1 500 d ), where h h represents the hours of sunlight, and d d is the day of the year. Use the equation to find how many hours of sunlight there are on September 24, the 267th day of the year. State the period of the function.
The equation P=20sin( 2πt )+100 P=20sin( 2πt )+100 models the blood pressure, P, P, where tt represents time in seconds. (a) Find the blood pressure after 15 seconds. (b) What are the maximum and minimum blood pressures?
The height of a piston, h, h, in inches, can be modeled by the equation y=2cosx+6, y=2cosx+6, where x x represents the crank angle. Find the height of the piston when the crank angle is 30°. 30°.
The height of a piston, h, h, in inches, can be modeled by the equation y=2cosx+5, y=2cosx+5, where x x represents the crank angle. Find the height of the piston when the crank angle is 55°. 55°.